Physica D 234 (2007) 131–149 www.elsevier.com/locate/physd

Polarization fields and phase space densities in storage rings: Stroboscopic averaging and the ergodic theorem✩

James A. Ellison∗, Klaus Heinemann

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, United States

Received 1 May 2007; received in revised form 6 July 2007; accepted 9 July 2007 Available online 14 July 2007

Communicated by C.K.R.T. Jones

Abstract

A class of orbital motions with volume preserving flows and with vector fields periodic in the “time” parameter θ is defined. Spin motion coupled to the orbital dynamics is then defined, resulting in a class of spin–orbit motions which are important for storage rings. Phase space densities and polarization fields are introduced. It is important, in the context of storage rings, to understand the behavior of periodic polarization fields and phase space densities. Due to the 2π time periodicity of the spin–orbit equations of motion the polarization field, taken at a sequence of increasing time values θ,θ 2π,θ 4π,... , gives a sequence of polarization fields, called the stroboscopic sequence. We show, by using the + + Birkhoff ergodic theorem, that under very general conditions the Cesaro` averages of that sequence converge almost everywhere on phase space to a polarization field which is 2π-periodic in time. This fulfills the main aim of this paper in that it demonstrates that the tracking algorithm for stroboscopic averaging, encoded in the program SPRINT and used in the study of spin motion in storage rings, is mathematically well-founded. The machinery developed is also shown to work for the stroboscopic average of phase space densities associated with the orbital dynamics. This yields a large family of periodic phase space densities and, as an example, a quite detailed analysis of the so-called betatron motion in a storage ring is presented. c 2007 Elsevier B.V. All rights reserved. ￿ Keywords: Periodic spin field; Dynamical systems; Ergodic theory; Stroboscopic averaging; Spin polarization

1. Introduction fields. The dimensions of a bunch are very small compared to the average radius of the ring, e.g., the proton–electron collider This paper explores certain mathematical matters relating HERA at DESY has a circumference of 4 miles and bunch to periodic particle and spin distributions in storage rings. dimension of a few millimeters. For the purposes of this paper Storage rings are large scale devices used to enable high energy we ignore the emission of electromagnetic radiation by the fundamental particles such as electrons, positrons, protons and particles and all collective effects, e.g., all interactions between nuclei to be brought to collision in order to study the most the particles and the effects of the electric and magnetic fields basic properties of matter. Descriptions of storage rings can be set up in the vacuum pipe by the particles themselves. Then found in standard text books and articles. See for example [21, particle motion is determined just by the Lorentz force [35] 27,37,43,44]. However, to summarize, the common feature of and a Hamiltonian can be assigned. Since we are dealing with a storage ring is that the electrically charged particles are storage rings we take the orbital motion to be bounded. confined to move in bunches on approximately circular orbits Information from analysis of observations from particle in a vacuum tube by combinations of electric and magnetic collisions in storage rings is much enhanced if the beams are “spin polarized”. Each electron, positron, proton or deuteron ✩ carries an intrinsic angular momentum called the “spin angular With support from DOE grant DE-FG02-99ER41104. momentum” and there is an associated 3-vector called the Corresponding author. Tel.: +1 505 2774110; fax: +1 505 2775505. ∗ “spin expectation value” s [24] which in this paper we simply E-mail addresses: [email protected] (J.A. Ellison), ￿ [email protected] (K. Heinemann). call the “spin”. This leads to the concept of a spin-valued

0167-2789/$ - see front matter c 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2007.07.007￿ 132 J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 function on phase space. The spin dynamics is governed the Cesaro` sequence converges for given (θ, z) and that it by the Thomas–Bargmann–Michel–Telegdi (T–BMT) equation is, to numerical accuracy, 2π-periodic in θ at each z. The ds/dt Ω(E(t), B(t), v(t)) s where E, B and v main accomplishments of this paper are precise definitions ￿ = ￿ ￿ ￿ ￿ ×￿ ￿ ￿ ￿ are respectively the electric and magnetic fields and the of the stroboscopic sequence, the Cesaro` sequence and the velocity [35]. Thus the particle spins precess in the electric stroboscopic average, the proof that the stroboscopic average and magnetic fields and their lengths s are constant. The is a 2π-periodic polarization field and our discussion of the |￿| quantity of interest to experimenters using a storage ring is the convergence properties of the Cesaro` sequence. An important beam polarization P￿, i.e., the average of the normalized spins. tool in our work is the Birkhoff ergodic theorem, called To obtain this we first average only over the spin degrees of henceforth the ergodic theorem. The rate of convergence of the freedom to get the local polarization P￿loc, i.e., the polarization Cesaro` sequence is only briefly treated here (see Section 3.4.2) at every point of phase space. Then the beam polarization is the but it is discussed, for the case of integrable orbital systems, phase space average of the local polarization. See Section 3.4.1. in [32]. The above mentioned spin-valued functions on phase space will A polarization field that is normalized and is 2π-periodic in be called polarization fields and will be defined in Section 3.2.1. θ at each z is now usually called an “invariant spin field”. This The local polarization is a polarization field. In a well set-up is an important object for systematizing spin dynamics. For storage ring the phase space density ρ in bunches is periodic example the invariant spin field plays a role in the calculation from turn to turn. The experimenters also desire that the local of the maximum attainable equilibrium beam polarization and polarization is periodic from turn to turn. It is thus of great the maximum attainable time averaged beam polarization at interest to find ways of calculating such one-turn periodic each θ [4]. The invariant spin field is also the starting point polarization fields. In order for the maximum information to for calculating the so-called amplitude dependent spin tune be extracted from experiments, it is desirable that the beam if the latter exists [3,4,7,17,18,33,39,42,45,46]. An invariant polarization be as large as possible. spin field is derived from a 2π-periodic polarization field via Although dynamical systems are often analyzed by taking stroboscopic averaging by simply normalizing the stroboscopic time as the independent variable, this is usually not convenient average to unity. for storage rings since there, the vacuum tube and the electric The discovery that invariant spin fields could be approxi- and magnetic guide fields have a fixed, approximately circular mated via stroboscopic averaging led to the creation of the com- spatial layout. It is therefore standard practice to begin analysis puter code SPRINT [32,33,42] as a way of computing invariant by constructing the curvilinear “closed orbit”, i.e., the orbit spin fields for very high energy where other algorithms would along which the particle motion is one-turn periodic. The fail or be impractical. SPRINT was then heavily used for com- equations of motion for the particles and their spins are then puting the invariant spin field in a study of the feasibility of at- transformed into forms in which the angular distance, θ, along taining high proton polarization in the electron–proton storage the closed orbit is taken as the independent variable and ring system, HERA [33,42]. Stroboscopic averaging can even particle positions are defined with respect to the closed orbit. handle exotic models where the invariant spin field is discon- The one-turn periodicity of the positions of the electric and tinuous in the orbit variables [9,10]. magnetic guide fields then becomes a 2π-periodicity in θ. The Although the numerical calculation of the invariant spin field new equations of motion can be derived from an appropriate by stroboscopic averaging represents a major advance in our Hamiltonian and the three pairs of canonical variables are ability to systematize spin motion in storage rings, there has combined into a vector z with six components. For example, been no detailed investigation of the Cesaro` sequence and its two of the pairs can describe transverse motion and one pair can convergence. Thus the bulk of the present paper consists of describe longitudinal (synchrotron) motion within a bunch. One introducing mathematical definitions and stating and proving of this latter pair quantifies the deviation of the particle energy related theorems and propositions. from the energy of a particle fixed at the center of a bunch and We begin in Section 2, with a discussion of the details the other describes the time delay with respect to that particle of the orbital dynamics necessary for the discussion of the (see e.g. [7]). With respect to the average radius of the closed spin dynamics. To pave the way for the application of the orbit and the nominal particle energy, the canonical position ergodic theorem to the spin dynamics, we apply it first to variable and the energy variable are very small. Although it the Cesaro` sequence of phase space densities defined by the would be natural to choose an appropriate name for the new evolution of ensembles of particles. We thereby show that independent variable, we simply refer to θ as the “time”. In the the stroboscopic average is a 2π-periodic phase space density following we will, for convenience, dispense with the symbol and that the Cesaro` sequence converges almost everywhere. “ ” over spin-like quantities. A brief summary of Section 2 is given in Section 2.4 and in → Arguably the most general and reliable method for Appendix A, the results of Section 2 are applied to the linear numerically calculating periodic polarization fields in storage one-degree-of-freedom time-periodic Hamiltonian case which rings involves “stroboscopic averaging” [32]. For this, one is important for storage rings. In Section 3, we turn our attention begins with an arbitrary polarization field, (θ, z), and to polarization fields. We first discuss the basics of spin motion S computes the average of (θ, z), (θ 2π, z), ..., (θ and polarization fields, and then set up the problem in a form S S + S + 2π(N 1), z) which as a function of the integer N forms for application of the ergodic theorem. Its application yields − the “Cesaro` sequence”. One usually finds in practice that our main results: the stroboscopic average of every polarization J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 133

field is a 2π-periodic polarization field and, if the phase space We denote the solution of the initial value problem (2.1) and is of finite measure, the Cesaro` sequence converges almost (2.2) by everywhere. This fulfills the main aim of this paper in that z(θ) ϕ(θ,θ z0),θ I (θ , z0), (2.3) it demonstrates that the tracking algorithm for stroboscopic = 0; ∈ 0 averaging, encoded in the program SPRINT and used in the where study of spin motion in storage rings, is mathematically well- founded. Finally, in Section 3.4, we discuss the physical D ϕ(θ,θ z0) f (θ,ϕ(θ,θ z0)), ϕ(θ ,θ z0) z0, 1 0; = 0; 0 0; = significance of our results. Section 4 is a summary and brief (2.4) discussion. 0 and where I (θ0, z ) is the maximal interval of existence at 0 2. The orbital system and Liouville densities (θ0, z ) (see Remark (1)). By the uniqueness of solutions, we have the basic identity The main purpose of this section is to set notation and ϕ(θ ,θ ϕ(θ ,θ z0)) ϕ(θ ,θ z0), (2.5) to acquaint the reader with stroboscopic sequences, Cesaro` 2 1; 1 0; = 2 0; sequences, stroboscopic averages and the ergodic theorem. and in addition the periodicity of f ensures that We prove that the stroboscopic averages are bounded, time- 1 0 0 0 periodic, phase space densities. This is important since ϕ(θ 2π,θ0 2π z ) ϕ(θ,θ0 z ),θ I (θ0, z ). (2.6) L + + ; = ; ∈ storage rings are only useful if the phase space density is Note that the periodicity of f ( , z) implies that if z( ) is a approximately periodic turn to turn. · · solution of (2.1) for all time then so is z( 2π). Note also that More specifically, in Section 2.1 we discuss the details of ·+ (2.5) holds subject to the constraint on the θ values imposed by the orbital motion necessary for our paper. In Section 2.2 we the maximum intervals of existence. introduce phase space densities, their stroboscopic sequences, Since f (θ, ) is divergence free, Cesaro` sequences, and stroboscopic averages. The ergodic · 0 0 theorem is stated in Section 2.3 and we illustrate its use in the det(D3ϕ(θ,θ0 z )) 1,θ I (θ0, z ). (2.7) simple context of phase space densities to construct periodic ; = ∈ phase space densities by stroboscopic averaging. Also, as a Our only assumption in addition to the properties of f is the concrete example, we discuss in Appendix A the linear one- existence of an open nonempty set U such that degree-of-freedom time-periodic Hamiltonian system which is ϕ(2π, 0 U) U. (2.8) important since the betatron motion in a storage ring is a special ; = subcase. Since we are dealing with a storage ring and neglect the The material after the statement of the ergodic theorem emission of radiation and collective effects this is a reasonable (Theorem 2.4) is not needed for Section 3, in particular assumption. In fact, integrable motion, which is often assumed Lemma 2.5 and Theorem 2.6 are not needed. However the in calculations, has this property (see Appendix A). We define 0 reader might find this material helpful as it is a simpler context Uθ ϕ(θ, 0 U) and from now on restrict the z in (2.2) to := ; for the application of the ergodic theorem. Uθ0 and take the domain of ϕ to be R L where L is the set of 0 × admissible initial conditions (θ0, z ), i.e.,

2.1. The orbital motion d 1 L (θ, z) R + z Uθ θ Uθ . (2.9) := { ∈ : ∈ }= { }× θ Consider the initial value problem ￿∈R From the invariance (2.8) of U under the period advance map z f (θ, z), (2.1) ϕ( π, ) (θ , 0) 0 ˙ = 2 0 , it is clear that I 0 z R if z Uθ0 and that 0 d ϕ(θ,θ ;·) 1 = ∈ z(θ0) z R , (2.2) 0 Uθ0 Uθ is a C -diffeomorphism onto Uθ . Note = ∈ ;· : →d d 1 that Uθ is open in R and L is open in R + . It is also clear that d 1 d 1 (θ, ) 0 where f R + R is of class C , f is divergence Uθ 2π Uθ and that (2.4)–(2.7) hold whenever z Uθ . : → · + = ∈ 0 free (Tr D2 f (θ, z) 0), f ( , z) is 2π-periodic and θ0 is We immediately draw an important conclusion from our [ ]= · 1 an arbitrary initial time. The choice that f is of class C is assumptions. Let (Uθ , θ ,µ ) be the measure space over the U d consistent with the fact that the electric and magnetic fields σ -algebra θ of Borel subsets of Uθ with Lebesgue measure U in storage rings are smooth. Throughout this paper Dk will µd , then it follows from (2.7) and the transformation theorem denote the derivative with respect to the k-th argument, be it for Lebesgue integrals (see for example [11, Section 19]) that scalar or vector. For simplicity the variable θ is called “time”. each ϕ(θ,θ ) is measure preserving, i.e., µ (ϕ(θ ,θ A)) 0;· d 0 ; = We want the results of this paper to apply to beam dynamics µd (A) for all A θ . In the following we define 0. in storage rings (see Introduction, Section 3.4, Summary and ∈ U U := U Remark. Appendix A). In this case, θ plays the role of the so-called 0 azimuthal variable and f is a Hamiltonian vector field 2π- (1) Let I (t0, z ) (α,β), then the standard continuation ≡ 1 d 1 periodic in θ with z being the vector of generalized position theorem for f in C (R + ) gives that either β or =∞ f z(t) as t β and similarly for α (see, e.g., [14, and momentum coordinates. However we do not assume that | |→∞ ↑ is generated by a Hamiltonian. Theorem 1.4] or [1,25]). ￿ 134 J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149

2.2. Liouville densities on phase space and their stroboscopic Note that ρg is 2π-periodic iff g satisfies the fixed point average equation g(z) g (ϕ(0, 2π z)) . (2.16) 2.2.1. Liouville densities and their basic properties = ; As motivation for our definition of Liouville densities, As we will see, (2.16) is the invariance property in the ergodic consider an ensemble of particles defined by a phase space theorem (see also Remark (3)). density ρ which evolves under the time flow defined by (2.1). More generally, we call ρg “2π-periodic in measure” From particle conservation we have if for every θ the equality: ρ (θ 2π, z) ρ (θ, z) g + = g holds for µd -almost every z. In that case there may be no ρ(0, z0)dµ (z0) ρ(θ, z)dµ (z) z such that ρ ( , z) is 2π-periodic, however the ‘average’ d d g · A = ϕ(θ,0 A) ρ (θ, z)F(θ, z)dµ (z) over a 2π-periodic ‘observable’ F ￿ ￿ ; Uθ g d 0 0 0 is a 2π-periodic function of θ. Note also that ρg is 2π-periodic ρ(θ,ϕ(θ, 0 z )) det(D3ϕ(θ, 0 z ))dµd (z ) ￿ = ; ; in measure iff g satisfies the fixed point equation (2.16) for µ - ￿A d almost every z. The optimal situation in a storage ring is for the 0 0 ρ(θ,ϕ(θ, 0 z ))dµd (z ), (2.10) physically relevant LD (see Section 3.4.1) to be 2π-periodic = A ; ￿ or, at least, 2π-periodic in measure. Note however that, in this where A . The first equality expresses particle conservation, ∈ U paper, those LD’s which are 2π-periodic in measure but not the second uses the transformation theorem for Lebesgue 2π-periodic play only a minor role. integrals and the third follows from (2.7). Since A is an arbitrary Liouville densities in accelerators are discussed in e.g. [20, element of it follows from (2.10) that, for µ -almost every U d 22,23], [27, Section 2.5] and their importance for polarized z Uθ , we have ρ(0, z) ρ (θ,ϕ(θ, 0 z)) so that ∈ = ; beams in storage rings will be further revealed in Section 3.4.1. ρ(θ, z) ρ (0,ϕ(0,θ z)) . (2.11) Remark. = ; 1 Since sets of measure zero are unimportant we are free to (2) If the LD ρg is C then it satisfies the first order PDE specify ρ(0, ) and define ρ everywhere by (2.11). Eq. (2.11) ∂ · ρ (ρ f (θ, z)) 0. (2.17) motivates our definition of a Liouville density (LD). ∂θ g +∇z · g = In the more general case where Tr D f 0, Eq. Definition 2.1 (Liouville Density). Let g U 0, ) be [ 2 ]￿= 1 : →[ ∞ (2.10) motivates the definition ρg(θ, z) g (ϕ(0,θ z)) exp a bounded function in (U, ,µd ). A function ρg L θ = ; L U : → ( Tr D f (θ ,ϕ(θ ,θ z)) dθ ) since the Wronskian 0, ) is called a “Liouville density” (LD) if − 0 [ 2 ￿ ￿ ; ] ￿ [ ∞ (θ, z0) det(D ϕ(θ, 0 z0)) satisfies D a W ￿ := 3 ; 1W = W ρg(θ, z) g (ϕ(0,θ z)) . (2.12) where a(θ, z0) Tr D f θ,ϕ(θ, 0 z0) . Thus this = ; := [ 2 ; ] more general definition of ρg satisfies (2.17) if f and g The function g will be called the ‘generator’ of ρg. ￿ ￿ ￿ are sufficiently smooth. The PDE (2.17) is often called For the definition of 1(U, ,µ ), see Remark (5). Clearly, the “Liouville equation” (or the “Vlasov equation” in the L U d every LD is a bounded function since every upper bound of g collective case). ￿ is an upper bound of ρ . Note that for z Uθ , ϕ(0,θ z) U g ∈ ; ∈ so that ρg is well defined. Moreover it is easy to check by (2.5) 2.2.2. The stroboscopic and Cesaro` sequences of a Liouville and (2.12), that at times θ1 and θ2 density In this section we define a stroboscopic sequence of the LD ρg(θ2, z) ρg (θ1,ϕ(θ1,θ2 z)) . (2.13) = ; ρg and the associated Cesaro` sequence, and derive properties of Since (θ , z) and (θ ,ϕ(θ ,θ z)) are on the same solution the Cesaro` sequence. 2 1 1 2; curve of (2.1) it follows from (2.13) that ρ is constant along g (Stroboscopic Sequence and Cesaro` Sequence). solution curves. Definition 2.2 Let ρ be an LD. The “stroboscopic sequence of ρ ” consists of Of course, by setting A U in (2.10) one obtains the g g = 1 the functions ρg( 2πn, ) L 0, ) where n 0, 1,.... obvious results for an LD ρg that ρg(θ, ) (Uθ , θ ,µd ) ·+ · : →[ ∞ = · ∈ L U Let ρ N L 0, ) be defined by and g : →[ ∞ N 1 N 1 − gdµ ρ (θ, z)dµ (z). (2.14) ρ (θ, z) ρg(θ 2πn, z) d g d g := N + U = Uθ n 0 ￿ ￿ ￿= N 1 It is convenient not to require these integrals to be 1. However 1 − we say that an LD ρ is “normalized” if gdµ 1 (then, g (ϕ(0,θ 2πn z)) , (2.18) g U d = = N + ; clearly, ρg(θ, z)dµd (z) 1). n 0 Uθ = ￿ ￿= We call an LD ρg “2π-periodic” iff ρg(θ 2π, z) , ,... ρ N ￿ + = where N 1 2 . The sequence g ∞N 1 is called the ρg(θ, z). Thus if ρg is 2π-periodic, (2.13) gives = {N } = “Cesaro` sequence of ρg” and the ρg are called “Cesaro` ρ (θ, z) ρ (θ,ϕ(θ,θ 2π z)). (2.15) averages of ρg”. ￿ g = g + ; J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 135

N In the special case when ρg is 2π-periodic, we have ρg Definition 2.3 (Stroboscopic Average). Let ρg be an LD. Then N = ρg. In the general case, we claim that ρg is an LD for each N. we call the function ρg L 0, ), defined by ρg(θ, z) N ˆ : →[ ∞ ˆ := We first argue that each element of the stroboscopic limN 1 g (z)ρg (θ, z), the “stroboscopic average of ρg”. →∞ Uθ sequence is an LD, i.e., that shifting the time by 2πn in ￿ the LD ρg gives an LD if n is an integer. In fact, since ρg(2πn, z) g(ϕ(0, 2πn z)) we see that g (ϕ(0, 2πn )) is Here 1A denotes the indicator function of the set A, i.e., = ; 1 ;· an 0, )-valued, bounded function in (U, ,µd ) and we 1A(z) 1 if z A and 0 otherwise; thus ρg(θ, z) is the [ ∞ L U = N ∈ ˆ have, by (2.5), (2.6) and (2.12), limit of ρg (θ, z) if it exists and zero otherwise. It follows from the definition that every LD ρg has a unique stroboscopic g (ϕ(0, 2πn ϕ(0,θ z))) average ρ which is a bounded and nonnegative function and ; ; ˆg g (ϕ(0, 2πn ϕ(2πn,θ 2πn z))) that ρg(θ, ) is θ - -measurable where is the σ -algebra = ; + ; ˆ · U R N R g (ϕ(0,θ 2πn z)) of Borel subsets of R. Note that ρg (θ, z) converges for all = + ; g (θ, z) L as N iff U U. In particular this is true ρg(θ 2πn, z), (2.19) ∈ →∞ 0 = = + if ρg is 2π-periodic. By (2.22) which proves that ρ ( 2πn, ) is an LD which is generated g ·+ · by g (ϕ(0, 2πn )) so that each element of the stroboscopic 1U g (z) 1U g (ϕ(0,θ z)) , (2.23) ;· θ = 0 ; sequence is an LD. It follows from (2.18) and (2.19) N and, since ρ N is an LD, we obtain that the function that, for every N, ρg is an LD which is generated by g N 1 N g ( )ρ N (θ, ) (1/N) − g (ϕ(0, 2πn )). Since ρ is an LD then by 1U z g z in Definition 2.3 is an LD. Thus the n 0 ;· g θ definition = stroboscopic average ρg is the limit of a sequence of LD’s. ￿ ˆ N N The main issues now are whether ρg is a 2π-periodic LD ρ (θ, z) ρ (0,ϕ(0,θ z)) . (2.20) g ˆ N g g and whether U is of full measure in Uθ . Since 1 g (z)ρ (θ, z) = ; θ Uθ g Having used (2.6) in (2.19) we see that the 2π-periodicity of is an LD we have N f ( , z) is essential in proving that each element ρg of the N N · 1U g (z)ρg (θ, z) 1U g (ϕ(0,θ z)) ρg (0,ϕ(0,θ z)) . stroboscopic sequence is an LD. θ = 0 ; ; N In addition ρ (θ, z)dµd (z) gdµd . To see this, Uθ g = U Then by taking N , we obtain note that by (2.14) and (2.18) →∞ ￿ ￿ ρg(θ, z) ρg (0,ϕ(0,θ z)) . (2.24) ρ N (θ, z)dµ (z) ρ N (0, )dµ ˆ =ˆ ; g d g d 1 Uθ = U · Thus if ρ (0, ) is in (U, ,µ ) then ρ is an LD. If U is of ￿ ￿ ˆg · L U d ˆg N 1 finite measure then this is true by the dominated convergence 1 − g (ϕ(0, 2πn )) dµd . theorem (see for example [11, Section 15]). = N ;· n 0 ￿U That ρ is a 2π-periodic LD and that U g is of full measure in ￿= ˆg θ Since ϕ(0, 2πn ) is measure preserving and ϕ(0, 2πn U) Uθ will be shown in Section 2.3 as a consequence of the ergodic ;· ; = U, the stated result follows. theorem. A subset of a measure space will be said to have full measure if its complement has measure zero. Thus, in the case 2.2.3. The stroboscopic average of a Liouville density of a finite measure space, a set is of full measure iff it has the We now discuss some convergence properties of the Cesaro` measure of the underlying measure space. g N sequence. Let U Uθ be the set on which ρ (θ, ) converges, θ ⊂ g · i.e., 2.3. Applying the ergodic theorem to the Cesaro` sequence of a g N Liouville density Uθ z Uθ lim ρg (θ, z) exists . (2.21) := { ∈ : N } →∞ g In this section we prove that the stroboscopic average of Clearly U θ and our goal is to show that this is large in θ ∈ U measure. every LD is a 2π-periodic LD. Our analysis will also show that N for the generally nonperiodic ρg in (2.12) its Cesaro` sequence From (2.20) ρg (0,ϕ(0,θ z)) converges on the same set, g ; g converges almost everywhere, i.e., that U is of full measure. thus ρ N (0, z) converges for z ϕ(0,θ U ) and so θ g ∈ ; θ These results follow easily from an application of the ergodic g g theorem, which we now state. Uθ ϕ(θ, 0 U0 ). (2.22) = ; An - -measurable map T on a measure space g M M 1 Since ϕ(0,θ ) is measure preserving, (2.22) gives µd (Uθ ) (M, , m) is said to be measure preserving if m(T − A) g ;· = M 1 = µd (U0 ). Note that since g is bounded, the sequence m(A) for all A in . A set A satisfying T − (A) A is N M ∈ M 1 = ρ (θ, z) ∞ is bounded. Thus the limit, if it exists, is said to be an invariant set and A T (A) A is { g }N 1 I := { ∈ M : − = } always a real= number. In this paper, the term “converges” the σ -algebra of T -invariant sets in . We now can state: without a qualifier will always mean pointwise converges. The M boundedness of the sequence also assures that for every pair Theorem 2.4 (Ergodic Theorem). Let T M M be : → (θ, z) a convergent subsequence exists but we will not use this a measure preserving map on the σ-finite measure space fact. (M, , m) and let b 1(M, , m). Then an element b of M ∈ L M ˇ 136 J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149

1(M, , m) 1(M, , m) exists such that, for m-almost It is easy to see that U g and ρ (0, ) 1(U, ,µ ) as L I ⊂ L M 0 ∈ I ˆg · ∈ L I d every z, follows. From (2.26) N 1 N 1 1 − 1 − n N n 1 lim b(T (z)) b(z), (2.25) ρg (0, T (z)) g(T + (z)) N N = ˇ = N →∞ n 0 n 0 ￿= ￿= 1 with b dm b dm. If m is a probability measure N N M ˇ M (g(T (z)) g(z)) ρg (0, z). (2.28) | | ≤ | | N 1 n 1 = N − + then the sequence (1/N) n −0 b(T ( )) converges in ￿ ￿ = · L g (convergence in the mean) to b and b is a version of the Since g is bounded it follows from (2.28) that z U0 iff ￿ ˇ ˇ g g ∈ conditional expectation E(b ). T (z) U . Thus U , therefore 1U g (T (z)) 1U g (z). ￿ ∈ 0 0 ∈ I 0 = 0 |I Hence by (2.28) There are many proofs of the ergodic theorem in the literature. 1 g (z) See for example [12,13,19]. Note that in Theorem 2.4 and in N U0 N 1U g (T (z))ρg (0, T (z)) (g(T (z)) g(z)) the following the phrase ‘m-almost every z’ always relates to 0 = N − (M, , m), not to (M, , m). N M I 1U g (z)ρg (0, z). (2.29) Remarks. + 0 Since ρ (0, ) is - -measurable, the - -measurability of (3) Since b in Theorem 2.4 is -measurable, b is invariant ˆg · U R I R ˇ I ˇ ρ (0, ) follows from the relation ρ (0, z) ρ (0, T (z)) for under T . In fact even the reverse holds, i.e., a - ˆg · ˆg =ˆg M all z and this equality holds for z U from (2.29) and measurable b is -measurable iff b(z) b(T (z)) [13, ∈ ˇ I ˇ = ˇ from Definition 2.3. Because ρ (0, ) is - -measurable and Section 6.4]. ˆg · I R since ρ (0, ) 1(U, ,µ ), it follows that ρ (0, ) (4) Note that in the σ-finite case b 1(M, , m) even ˆg · ∈ L U d ˆg · ∈ ˇ ∈ L M 1(U, ,µ ). though the convergence may not be 1. L I d L Also, we have by (2.27) and Theorem 2.4 that ρg(0, )dµd (5) We define for a measure space (M, , m) the real vector U ˆ · M ρg(0, ) dµd g dµd g dµd gdµd . space p(M, , m) in the same way as in [11, Section 14]. = U |ˆ · | = U |ˇ| ≤ U | | =￿ U L M Now let µd (U) 1, i.e., let (U, ,µd ) be a probability Then b p(M, , m) means that b is a real-valued ￿ =￿ ￿ U ￿ ∈ L M space. Thus g E(g ) and, by (2.27), - -measurable function which is p-fold m-integrable, ˇ = |I M R i.e. b pdm < where 1 p < . Note that b is a M | | ∞ ≤ ∞ ρ (0, )dµ gdµ gdµ , (2.30) function, not an equivalence class of functions. For brevity ˆg · d = ˇ d = d ￿ A A A we sometimes call a function “measurable” if it is clear ￿ ￿ ￿ for A . Since ρ (0, ) is - -measurable, we conclude which σ -algebras are involved. ∈ I ˆg · I R that ρ (0, ) E(g ). If µ (U) is finite but arbitrary then (6) Recall that the two defining properties of E(b ) ˆg · = |I d |I : we can repeat this argument by using the probability measure M R are that E(b ) is - -measurable and that → |I I R µd /µd (U) and again obtain (2.30). Thus (2.30) holds whenever A E(b )dm A bdm for all A . ￿ |I = ∈ I µd (U) is finite. We￿ now apply the￿ ergodic theorem to ρg in (2.12).We We have thus proved the following lemma about the θ 0 = define T ϕ(0, 2π ) so that T U U is a measure section: := ;· : → preserving C1-diffeomorphism onto U. In addition we take Lemma 2.5. Let ρg be an LD and ρg(0, ) be defined as in M U, , m to be the Lebesgue measure, and we set g ˆ · = M = U Definition 2.3. Then U A ϕ(2π, 0 A) b g so that b g. Thus A ϕ(2π, 0 A) A . 0 ∈ I ={ ∈ U : ; = = ˇ =ˇ I ={ ∈ U : ; n = } A and it is of full measure in U. Moreover ρ (0, ) is in Note that, by (2.5) and (2.6), ϕ(0, 2πn ) T . Thus, by } ˆg · ;· = 1(U, ,µ ) 1(U, ,µ ) and satisfies the condition (2.18), L I d ⊂ L U d U ρg(0, )dµd U gdµd and the invariance property N 1 ˆ · ≤ N 1 − ρg(0, z) ρg(0,ϕ(0, 2π z)). ρg (0, z) g(ϕ(0, 2πn z)) ￿ˆ =ˆ ￿ ; = ; If µd (U) 1 then ρg(0, ) E(g ) and U ρg(0, )dµd N n 0 = ˆ · = |I ˆ · = ￿= U gdµd . The latter equality holds whenever µd (U) is finite. N 1 ￿ 1 − ￿ g(T n(z)). (2.26) ￿ = N n 0 Note that, having used (2.6), we see that the 2π-periodicity ￿= of f ( , z) is essential for our proof of Lemma 2.5. N 1 n · By (2.26) (1/N) − g(T (z)) converges as N iff With Lemma 2.5 we easily obtain: g n 0 g →∞ z U . Hence, by Theorem= 2.4, U is of full measure. Thus ∈ 0 ￿ 0 applying (2.26) and Theorem 2.4 again, we obtain, for µ - Theorem 2.6. Let ρg be an LD. Then its stroboscopic average d g almost every z U ρg is a 2π-periodic LD, Uθ is of full measure in Uθ and ∈ ˆ g g Uθ 2π Uθ . Furthermore ρ (0, ) g. (2.27) + = ˆg · =ˇ 1 1 1 Since g (U, ,µd ), (U, ,µd ) (U, ,µd ) ρg(θ, z)dµd (z) gdµd . (2.31) ˇ ∈ L I L I ⊂ L U U ˆ ≤ U and since ρ (0, ) is - -measurable, we see by (2.27) that ￿ θ ￿ ˆg · U R ρ (0, ) 1(U, ,µ ). Equality holds if µd (U)< . ˆg · ∈ L U d ∞ J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 137

Recall that in Lemma 2.5 and Theorem 2.6 µd is the The long time stability of 2π-periodic Liouville densities is Lebesgue measure. an important issue for storage rings but we have not investigated this. Proof of Theorem 2.6. As pointed out in Section 2.2.3, the In Appendix A, we apply our theory to the important stroboscopic average ρ is a bounded and nonnegative function. ˆg case of linearized particle motion in a storage ring, the so- Because, by Lemma 2.5, ρ (0, ) 1(U, ,µ ) and since ˆg · ∈ L U d called integrable betatron motion in one degree of freedom. ρ 0, it follows from (2.24) that ρ is an LD. g g However, it is important to note that our theory does not require ˆ To≥ show that the LD ρ is 2π-periodicˆ we first note that since ˆg integrability. N N ρg (θ 2π, z) ρg (θ, z) + − 3. The spin–orbit system and polarization fields 1 ρg(θ 2π N, z) ρg(θ, z) , (2.32) = N + − We remind the reader of the brief overview of this section in and since￿ ρ is bounded, we have ￿z U g iff z U g , the last paragraph of the Introduction. This section runs parallel g ∈ θ ∈ θ 2π g g + to Section 2 in the main. After introducing the spin–orbit i.e., Uθ 2π Uθ . Thus we conclude from (2.32) that + = motion in Section 3.1 we turn our attention to polarization N N 1 g (z)ρ (θ 2π, z) 1 g (z)ρ (θ, z) Uθ 2π g Uθ g fields. Because Liouville densities are simpler than polarization + + − 1 g (z) fields, the proof of Lemma 2.5 shows in a nutshell how the Uθ ρg(θ 2π N, z) ρg(θ, z) . ergodic theorem applies to stroboscopic sequences. Therefore = N + − the proof of Lemma 2.5 may help the reader to go through By taking the￿ limit as N we obtain￿ ρ (θ, z) ρ (θ →∞ ˆg =ˆg + the more complicated application of the ergodic theorem to 2π, z). g the Cesaro` sequences of polarization fields (see the proof of Because by Lemma 2.5, U0 is of full measure in U and Lemma 3.6). since ϕ(0,θ ) is measure preserving, U g is, by (2.22), of full ;· θ measure in Uθ . 3.1. The spin–orbit motion It follows from Lemma 2.5 that ρ (0, )dµ U ˆg · d ≤ gdµ . Since ρ is an LD, (2.14) gives ρ (0, )dµ Now that we have tackled the purely orbital system and U d ˆg ￿ U ˆg · d = ρg(θ, z)dµd (z) and (2.31) follows. For finite measure µd illustrated the basic ideas, we are ready to apply our techniques ￿Uθ ˆ ￿ we have from Lemma 2.5 that ρ (0, )dµ gdµ . to the spin–orbit system, which is more complicated. As we ￿ U ˆg · d = U d Therefore equality holds in (2.31). ￿ explained in Section 1, spin motion is governed by the T–BMT ￿ ￿ equation [35]. The equations of the spin–orbit motion can then 2.4. Summary and discussion be written as 0 d z f (θ, z), z(θ0) z R , (3.1) An initial ensemble of particles, g, is given and ˙ = = ∈ ϕ 0 3 assumed to evolve according to the flow of f by s (θ, z)s, s(θ0) s R , (3.2) ρg(θ, z) g (ϕ(0,θ z)). An open nonempty set U which ˙ = A = ∈ = ; where the 3 3 matrix which represents the rotation vector is invariant under the flow is assumed to exist. For each × A π Ω in the T–BMT equation, is skew-symmetric ( T ) and g, we have constructed a 2 -periodic Liouville density ￿ A =−A N N real. Moreover, ( , z) is 2π-periodic and d 1 6 is ρg(θ, z) limN 1 g (z)ρg (θ, z) where ρg (θ, z) R + R ˆ = →∞ Uθ = 1 A · A1 : → N 1 g of class C . The choice that is of class C is consistent with (1/N) − ρ (θ 2πn, z) (see (2.18)) and where U A n 0 g + θ = the fact that the electric and magnetic fields in storage rings are ϕ(θ, 0 U g=) is of full measure as we proved in Theorem 2.6 ;￿0 smooth. using the ergodic theorem, Theorem 2.4. Eq. (3.1) describes the same orbital motion as in Section 2 so The definition of the flow ϕ, the invariant set U and the 0 0 that we continue to restrict z to Uθ . Thus z(θ) ϕ(θ,θ z ) 0 = 0; Liouville density are given in (2.4) and (2.8) and Definition 2.1 for all time and (2.4)–(2.7) hold; also U ϕ(2π, 0 U). respectively. The stroboscopic average ρg of ρg is defined in = ; gˆ N Because of the linearity of (3.2) in s the solutions of (3.1) and Definition 2.3 and the convergence set Uθ of ρg (θ, ) is defined 0 · (3.2) exist for all time since z Uθ0 . in (2.21). The main results of Section 2 are Lemma 2.5 and ∈ From now on we restrict the s0 in (3.2) to V s R3 Theorem 2.6 and the main tool is the ergodic theorem. In := { ∈ : 1 s < a with a > 0, where is the Euclidean norm. Let Lemma 2.5 we have constructed an (U, ,µd ) fixed point | | } |·| L I w z and f (θ, w) f (θ, z) , then from (3.1) and (3.2) solution of (2.16), namely ρ (0, ); in fact, the fixed point s so (θ, z)s ˆg · = = A condition is the invariance property which is equivalent to - we obtain￿ ￿ ￿ ￿ I measurability. Theorem 2.6 is proved using Lemma 2.5 and 0 w fso(θ, w), w(θ0) w Uθ V. (3.3) yields the following. If µ (U)< and g is normalized then ˙ = = ∈ 0 × d ∞ ρ is normalized. If g is normalized and µ (U) , the We denote the solution of (3.3) by w(θ) W(θ,θ w0) where ˆg d =∞ = 0; strict inequality may hold in (2.31) (see for example [40,41]) the domain of W is R L V where L is defined in (2.9). Note × × in which case ρ is not normalized. In fact ρ (θ, ) could be that the periodicity of f implies that if w( ) is a solution of ˆg ˆg · so · zero almost everywhere, however, if not, it can be normalized (3.3) then so is w( 2π). In analogy with (2.5) and (2.6) we ·+ by simply multiplying by a constant. have 138 J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149

W (θ ,θ W(θ ,θ w)) W(θ ,θ w), (3.4) evolve to z(θ) ϕ(θ, 0 z0) and s(θ) Ψ(θ, 0 z0)G(z0) at 3 2; 2 1; = 3 1; = ; = ; time θ. Let (θ, z) be the spin which is at the point z z(θ) W(θ2 2π,θ1 2π w) W(θ2,θ1 w). (3.5) SG = + + ; = ; at time θ. Then Since fso(θ, ) is divergence free, · (θ, z) Ψ (θ,0 ϕ(0,θ z)) G (ϕ(0,θ z)) . (3.14) SG = ; ; ; det(D W(θ,θ w0)) 1. (3.6) 3 0; = Definition 3.1 (Polarization Field). We call a function From our assumptions on f,ϕ and and noting that each SG : d 3 A 1 L 3 Uθ V is open in R + , it is clear that W(θ,θ0 ) is a C - R a “polarization field”, if it satisfies (3.14) where × ;· → 3 diffeomorphism from Uθ V onto Uθ V . We take as our G b( , ). ￿ 0 × × ∈ M U R ( , ,µ µ ) 3 3 measure space Uθ V θ d 3 where is the Here ( θ , ) is the set of bounded θ - -measurable × U ⊗ V × V Mb U R U R collection of Borel subsets of V , θ is the product σ- functions where 3 is the σ-algebra of Borel subsets of R3 and U ⊗ V R algebra of θ and and µ3 is Lebesgue measure restricted as before . The function G will be called the ‘generator’ U V U0 = U to . It follows from (3.6) and the transformation theorem for of G . Note that by (3.14) and by the fact that ϕ(0,θ ) and the V (θ,θ ) S ;· Lebesgue integrals that each W 0 is measure preserving. nine components of Ψ(θ, 0 ) are measurable functions, G is ;· ;· 3 S Furthermore, U V is an invariant set in the sense that U V a bounded function and (θ, ) is in ( θ , ). Of course, × × = SG · Mb U R W(2π, 0 U V ). (0, ) G. Using (2.5) and (3.11), the relation between the ; × SG · = Inserting the orbital dynamics into (3.2) yields values of the polarization field at times θ1 and θ2 is given by 0 0 s (θ,ϕ(θ,θ0 z ))s, s(θ0) s . (3.7) (θ , z) Ψ(θ ,θ ϕ(θ ,θ z)) (θ ,ϕ(θ ,θ z)). (3.15) ˙ = A ; = SG 2 = 2 1; 1 2; SG 1 1 2; Because of the smoothness of and ϕ and the linearity of (3.7), The polarization field has two very different properties: the A solutions exist for all time and can be written as “dynamical” condition (3.14) and “regularity” conditions. In contrast to the dynamical condition, the regularity conditions s(θ) Ψ(θ,θ z0)s0, (3.8) = 0; are to a certain extent a matter of convenience. For example where Ψ is a function on R L, called the “spin transfer in this paper we choose weak conditions, namely that G × 3 ∈ matrix”, and is defined by b( , ), because they serve us well when we come M U R to stroboscopic averaging in later sections. However, one 0 0 0 D1Ψ(θ,θ0 z ) (θ,ϕ(θ,θ0 z ))Ψ(θ,θ0 z ), (3.9) sometimes drops the condition that G is bounded and on ; = A ; ; 0 other occasions one assumes that G is of class Ck for some Ψ(θ0,θ0 z ) I3 3. (3.10) ; = × nonnegative integer k. In Section 3.4.4 we discuss what happens 1 Note that Ψ(θ,θ0 ) is of class C on Uθ0 . Since is real if one drops the regularity conditions. ;· T A skew-symmetric, Ψ is real, ΨΨ I3 3 and det(Ψ) 1. It is desirable that, in a storage ring with a polarized beam, = × = Thus Ψ SO(3) and s(θ) s0 for all θ as mentioned the physically relevant polarization fields (see Section 3.4.1) be ∈ | |=| | 0 0 ϕ(θ,θ0 z ) in Section 1. It is clear that W(θ,θ w ) ; . It 2π-periodic, i.e., G (θ 2π, z) G (θ, z) or, at least, 2π- 0 Ψ(θ,θ z0)s0 ; = 0; S + = S 0 periodic in measure. We say that a polarization field G is “2π- follows from (3.4) and (3.5) that for all z Uθ￿ ￿ S ∈ 0 periodic in measure” if for every θ the equality (θ 2π, z) SG + = 0 0 0 (θ, z) holds for µ -almost every z in Uθ . Note that if is Ψ(θ2,θ1 ϕ(θ1,θ0 z ))Ψ(θ1,θ0 z ) Ψ(θ2,θ0 z ), (3.11) SG d SG ; ; ; = ; 2π-periodic then (3.15) gives Ψ(θ 2π,θ 2π z0) Ψ(θ,θ z0). (3.12) + 0 + ; = 0; G (θ, z) Ψ (θ 2π,θ ϕ(θ,θ 2π z)) Taking θ2 θ0 in (3.11) we obtain S = + ; + ; = G (θ,ϕ(θ,θ 2π z)) . (3.16) 1 0 0 T × S + ; Ψ − (θ ,θ z ) (Ψ(θ ,θ z )) 1 0; = 1 0; Spin fields and invariant spin fields play an important role in the 0 Ψ(θ0,θ1 ϕ(θ1,θ0 z )). (3.13) theory of polarized beam physics and we now define them. = ; ; (θ π ,θ ) It is interesting to note that the Ψ 0 2 n 0 form a Definition 3.2 (Spin Field and Invariant Spin Field). A ( ) + ;· measurable SO 3 -cocycle over the orbital Z-action on Uθ0 polarization field with (θ, z) 1 is called a “spin SG |SG |= where n varies over the integers [28,29]. This is reflected in field”. A spin field is called an “invariant spin field” if it is 2π- the identity (3.11). periodic in measure. ￿ 3.2. Polarization fields and their stroboscopic sequence Thus a polarization field is a spin field iff G(z) SG | |= 1. Clearly 2π-periodic spin fields are invariant spin fields as 3.2.1. Polarization fields and their basic properties mentioned already in Section 1 and, in this paper, the emphasis In this section we will outline the basic properties of is on 2π-periodic invariant spin fields. polarization fields mentioned in Section 1 and we will define The concept of the invariant spin field was introduced into them rigorously below. Their significance in beam physics is polarized beam physics in the 1970s [17,18], but at that time discussed in Section 3.4. Consider an initial assignment of spins mathematical conditions like the regularity conditions were G U R3, i.e., a spin attached to every phase space point. not used explicitly. Moreover, the concept arose within a : → Under the flow of (3.3), the point z0 and its attached spin G(z0) Hamiltonian integrable description of spin–orbit motion. J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 139

If is a 2π-periodic polarization field, then from (3.14) G SG Definition 3.3 (Stroboscopic Sequence and Cesaro` Sequence). must satisfy the basic fixed point or functional equation Let G be a polarization field. The “stroboscopic sequence of S 3 ( ) Ψ ( π, ϕ( , π )) (ϕ( , π )) . G ” consists of the functions G ( 2πn, ) L R where G z 2 0 0 2 z G 0 2 z (3.17) S N S3 ·+ · : → = ; ; ; n 0, 1,.... Let G L R be defined by Recall the analogous fixed point equation (2.16) and note the = S : → N 1 similarity to homological equations in dynamical systems. This 1 − N (θ, z) (θ 2πn, z) important equation was, at least in the context of polarized SG := SG + N n 0 beam physics, probably first exploited by Yokoya and by ￿= N 1 π 1 − Balandin and Golubeva [2,47]. 2 -periodic polarization fields Ψ (θ 2πn, 0 ϕ(0,θ 2πn z)) are also discussed in [7,8,17,18,33,39,42,45,48]. Note that = + ; + ; N n 0 (3.17) is also a sufficient condition for the 2π-periodicity of ￿= SG G (ϕ(0,θ 2πn z)) , (3.20) since × + ; , ,... N (θ π, ) (θ π, π ϕ( π,θ π )) where N 1 2 . The sequence G ∞N 1 is called the G 2 z Ψ 2 2 2 2 z = {NS } = S + = + ; + ; “Cesaro` sequence of G ” and the G are called “Cesaro` G (2π,ϕ(2π,θ 2π z)) S S × S + ; averages of G ”. ￿ Ψ(θ, 0 ϕ(0,θ z)) G (2π,ϕ(0,θ z)) S = ; ; S ; Of course, in the special case when G is 2π-periodic, Ψ(θ, 0 ϕ(0,θ z)) N S = ; ; we have G . In the general case we claim that SG = S Ψ(2π, 0 ϕ(0, 2π ϕ(0,θ z))) N is a polarization field for each N. We first argue that × ; ; ; SG G(ϕ(0, 2π ϕ(0,θ z))) each element of the stroboscopic sequence is a polarization × ; ; Ψ(θ, 0 ϕ(0,θ z))G(ϕ(0,θ z)) field, i.e., that shifting the time by 2πn in the polarization = ; ; ; field G gives a polarization field if n is an integer. In fact, G (θ, z), (3.18) S = S Ψ (2πn, 0 ϕ(0, 2πn )) G (ϕ(0, 2πn )) is in ( , 3) ; ;· ;· Mb U R where we used (3.15) in the first equality, (2.6) and (3.12) in and we have, by (2.5), (2.6), (3.11), (3.12) and (3.14), the second, (3.14) in the third and fifth equalities and (3.17) in (θ, ϕ( ,θ )) ( π , ϕ( , π ϕ( ,θ ))) the fourth equality. We conclude that a polarization field is Ψ 0 0 z Ψ 2 n 0 0 2 n 0 z SG ; ; ; ; ; 2π-periodic iff G satisfies (3.17) for z U. It is also easy to G(ϕ(0, 2πn ϕ(0,θ z))) Ψ(θ, 0 ϕ(0,θ z)) ∈ ; ; = ; ; show that a polarization field G is 2π-periodic in measure iff Ψ(2πn, 0 ϕ(0,θ 2πn z))G(ϕ(0,θ 2πn z)) S × ; + ; + ; G satisfies (3.17) for µd -almost z in U. 1 Ψ(θ 2πn, 2πn ϕ(0,θ z)) If the polarization field G is of class C then, due to (2.4), = + ; ; S Ψ(2πn, 0 ϕ(0,θ 2πn z))G(ϕ(0,θ 2πn z)) (3.9) and (3.14), it satisfies the first order PDE × ; + ; + ; Ψ(θ 2πn, 0 ϕ(0,θ 2πn z))G(ϕ(0,θ 2πn z)) D1 G (D2 G ) f (θ, z) (θ, z) G . = + ; + ; + ; S + S = A S G (θ 2πn, z), (3.21) The following remarks will be useful. = S + which proves that ( 2πn, ) is a polarization field which Remarks. SG ·+ · is generated by Ψ (2πn, 0 ϕ(0, 2πn )) G (ϕ(0, 2πn )). (7) Let be a polarization field and let, for every z U, ; ;· ;· SG ∈ Thus each element of the stroboscopic sequence is a G(z) 0. Since ￿= polarization field. It follows from (3.20) and (3.21) that, N G (θ, z) G (ϕ(0,θ z)) , (3.19) for every N, is a polarization field generated by |S |=| ; | SG 3 N 1 we have G (θ, z) 0. Since G/ G is in b( , ) and (1/N) n −0 Ψ (2πn, 0 ϕ(0, 2πn )) G (ϕ(0, 2πn )). Since S ￿= | | M U R N = ; ;· ;· since by (3.14) and (3.19) is a polarization field then by definition SG ￿ G (θ, z) G (ϕ(0,θ z)) S Ψ (θ,0 ϕ(0,θ z)) ; , N (θ, z) Ψ (θ,0 ϕ(0,θ z)) N (0,ϕ(0,θ z)) . (3.22) (θ, z) = ; ; G (ϕ(0,θ z)) SG = ; ; SG ; |SG | | ; | we obtain that / is a spin field generated by G/ G . SG |SG | | | If is 2π-periodic then / is an invariant spin field. 3.2.3. The stroboscopic average of a polarization field SG SG |SG | (8) If G is a polarization field and ρg is an LD (recall We now discuss some convergence properties of the Cesaro` S G N Section 2.2.1) then ρ is a polarization field. This can sequence. Let U Uθ be the set on which (θ, ) gSG ˜θ ⊂ SG · be demonstrated in a way similarly to that in Remark (7). converges, i.e., Thus if a 2π-periodic spin field and a large number of 2π- G N periodic LD’s exists, one has a large number of 2π-periodic Uθ z Uθ lim G (θ, z) exists , (3.23) ˜ := { ∈ : N S } polarization fields. ￿ →∞ G Clearly U θ and our goal is to show that this is large in ˜θ ∈ U 3.2.2. The stroboscopic and Cesaro` sequences of a polariza- measure. tion field N From (3.22) G (0,ϕ(0,θ z)) converges on the same set, In this section we define a stroboscopic sequence of the N S ; G thus (0, z) converges for z ϕ(0,θ Uθ ) and so polarization field and the associated Cesaro` sequence, and SG ∈ ; ˜ SG derive properties of the Cesaro` sequence. U G ϕ(θ, 0 U G ). (3.24) ˜θ = ; ˜0 140 J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149

g Since ϕ(0,θ ) is measure preserving, (3.24) gives µd (Uθ ) G (θ 2π, z) G (θ, z) so that the stroboscopic average is g ;· = Sˆ + = Sˆ µd (U0 ). Note that since G is bounded, the sequence 2π-periodic. We have thus proved: N G (θ, z) ∞N 1 is bounded. Thus the limit, if it exists, is always {S } = Theorem 3.5. Let be a polarization field. Then its in R3. Furthermore since SG stroboscopic average is a 2π-periodic polarization field. SˆG N (θ 2π, z) N (θ, z) Furthermore U G U G and U G A G G ˜θ 2π = ˜θ ˜0 ∈ I ={ ∈ U : S + − S ϕ(2π, 0 A) A+. 1 ; = } ￿ ( G (θ 2π N, z) G (θ, z)) , (3.25) = N S + − S Having used (2.6) and (3.12) in (3.21) we see that the 2π- periodicity of f ( , z) and ( , z) is essential in our proof. Note G G · A · and since G is bounded, we have z U˜θ iff z U˜θ 2π , also that we did not need the ergodic theorem in our proof of G S G ∈ ∈ + i.e., Uθ 2π Uθ . The latter equality and (3.24) imply that Theorem 3.5 since for a polarization field we do not have to ˜ + = ˜ U G ϕ(2π, 0 U G ), i.e. U G . demonstrate that its components are integrable. ˜0 = ; ˜0 ˜0 ∈ I Remark. Definition 3.4 (Stroboscopic Average). Let G be a polariza- (9) It follows from Theorem 3.5 that (0, z) satisfies the fixed S 3 ˆG tion field. Then we call the function G L R , defined by S N Sˆ : → point Eq. (3.17), i.e., G (θ, z) limN 1 G (z) (θ, z), the “stroboscopic aver- ˆ Uθ G S := →∞ ˜ S G (0, z) Ψ (2π, 0 ϕ(0, 2π z)) G (0,ϕ(0, 2π z)) . age of G ”. ￿ Sˆ = ; ; Sˆ ; S (3.29) It follows from the definition that G (θ, z) is the limit of 3 Sˆ Recalling Remark (3), G (0, ) is - -measurable iff N (θ, z) if it exists and zero otherwise. Furthermore every Sˆ · I R SG G (0, ) G (0,ϕ(0, 2π )) . (3.30) polarization field G has a unique stroboscopic average ˆG Sˆ · = Sˆ ;· S (θ, ) ( , S3) which is a bounded function and ˆG is in b θ . In the trivial case where 0 (hence Ψ I3 3), N S · M U R A = 3 = × We also note that G (θ, z) converges for all (θ, z) L as (3.29) implies (3.30) and thus G (0, ) is - -measurable. G S ∈ Sˆ · I R N iff U0 U. In particular this is true if G is 2π- However, if 0 then Ψ I3 3 so that (3.30) does not →∞ ˜ = S A ￿= ￿= 3× periodic. It is also clear that U G U if has no zeros. We hold whence G (0, ) is not - -measurable. ￿ ˜0 = SˆG Sˆ · I R note by (3.24) that 3.3. Applying the ergodic theorem to the Cesaro` sequence of a polarization field 1U G (z) 1U G (ϕ(0,θ z)) , (3.26) ˜θ = ˜0 ; N With Theorem 3.5 we see that the stroboscopic average of and, since G is a polarization field, we see that the function N S a polarization field has the important property that it is a 2π- 1 G (z) (θ, z) in Definition 3.4 is a polarization field. Thus U˜θ SG periodic polarization field. In this section we use the ergodic the stroboscopic average is the limit of a sequence of SˆG theorem to prove that the Cesaro` sequence converges (at least) polarization fields. almost everywhere. In particular, we will prove Theorem 3.9 In Section 3.3 it will be shown, as a consequence of the which states that if U has finite measure then, for every θ, U G is G ˜0 ergodic theorem, that U is of full measure in Uθ if U has finite N ˜θ of full measure. It follows that the sequence G (θ, z) converges measure. We now show that is a 2π-periodic polarization S ˆG for µd -almost every z to G (θ, z) as N . N S Sˆ →∞ N field. Since 1U G (z) G (θ, z) is a polarization field we have By (3.24) the convergence of a Cesaro` sequence is ˜θ S SG determined by the convergence of the sequence N (0, ). N SG · 1 G (z) G (θ, z) U˜θ Therefore the following preparatory lemma studies the S N N sequence G (0, ) in detail. With Ψ (θ,0 ϕ(0,θ z)) 1U G (ϕ(0,θ z)) G (0,ϕ(0,θ z)) . S · = ; ; ˜0 ; S ; G (z) N (0, z) By taking N , we obtain N := SG →∞ N 1 1 − Ψ(2πn, 0 ϕ(0, 2πn z))G(ϕ(0, 2πn z)), G (θ, z) Ψ (θ,0 ϕ(0,θ z)) G (0,ϕ(0,θ z)) . (3.27) = ; ; ; Sˆ = ; ; Sˆ ; N n 0 ￿= Thus and since (0, ) ( , 3) we find that is a (3.31) SˆG · ∈ Mb U R SˆG polarization field. To show that it is 2π-periodic, we conclude we obtain from (3.25) that 3 N N Lemma 3.6. Let U be of finite measure and G in b( , ). 1 G (z) G (θ 2π, z) 1 G (z) G (θ, z) G M U R U˜θ 2π U˜θ Then U is of full measure in U, i.e., for µd -almost every z, + S + − S ˜0 1U G (z) ˜θ ( (θ 2π N, z) (θ, z)) , (3.28) lim G N (z) G (0, ). (3.32) G G N = Sˆ · = N S + − S →∞ where we also used the fact that U G U G . Taking N Proof of Lemma 3.6. The lemma will be proved with the aid ˜θ 2π = ˜θ →∞ in (3.28) and using the fact that + is bounded we find that of two propositions. SG J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 141

We begin by defining the measure preserving period advance Proof of Proposition 3.8. Clearly 1 s (z) 1 (z, s) so that F = F map associated with (3.3). The period advance map U Fs . Thus P : × ∈ U V U V in the θ 0 section and its inverse are given by → × = s µd (F ) 1Fs (z)dµd (z) 1F (z, s)dµd (z), (3.35) 1 = = (w) W(2π, 0 w), − (w) W(0, 2π w). (3.33) ￿U ￿U P = ; P = ; µ ( s) 1 1 and d F is a - -measurable and bounded function of Of course, and − are measure preserving C - V R s P P s. By (3.35) and the Fubini theorem µd (F )dµ3(s) diffeomorphisms onto U V . This follows from Section 3.1 V = × U V 1F d(µd µ3) (µd µ3)(F) µd (U)µ3(V ) and since U2π U. × × = × ￿ = s = = µd (U)dµ3. Therefore (µd (U) µd (F ))dµ3(s) ￿V V − = The key to our proof is the application of the ergodic theorem 0. Since the integrand is nonnegative, a) follows (see for (Theorem 2.4) to the function b U V R defined by ￿ ￿ : × → example [11, Section 13]). If (b) were not true then all s in VF would lie in a plane in 3 and we would obtain the contradiction b(w) sTG(z). (3.34) R := that µ3(VF ) 0. ￿ 3 = Since U is of finite measure and G b( , ) we have 1 ∈ M U R We now complete the proof of Lemma 3.6. The set b (U V, ,µd µ3). With this b and with ∈ L 1 × U ⊗ V × E of Proposition 3.7 is of full measure so we can apply T , , m µ µ , Theorem 2.4 can 1 2 3 = P− M = U ⊗ V = d × 3 Proposition 3.8 to it. Choose s , s , s as in Proposition 3.8(b) be used to prove the first proposition, namely Proposition 3.7 1 s1 s2 s3 1 and let U E E E . Then µd (U ) µd (U). If below. 1 := k ∩ ∩ = k T z U then (z, s ) E and, by Proposition 3.7, (s ) G N (z) The b in (3.34) is not an obvious choice but we need a scalar ∈ k ∈ 1 2 3 T T converges to b(z, s ) as N . Let A s , s , s then function of w and this is a simple one. Furthermore s G (θ, z) ˇ →∞ =[ ] S 1 T is conserved along solutions of (3.3) as is easily seen using (s ) G N (z) 0 0 0 1 1 2 T (3.14) and the relations z ϕ(θ, 0 z ), s Ψ(θ,θ0 z )s G N (z) A− AGN (z) A− (s ) G N (z) , = ; = ; = =  3 T  and it has the formal structure of a “spin action” (see e.g. [7, (s ) G N (z) 45]). However the only justification which matters is that the   ( , 1) ansatz (3.34) works. bˇ z s 1 1 ( , 2) 1 which converges, when z U , to A− bˇ z s . Since U is ∈ ( , 3) ￿bˇ z s ￿ Proposition 3.7. There exists a function b in 1(U V, ( ) µ ˇ L × U ⊗ of full measure, we have shown that G N z converges for d - ,µd µ3) and a set E of full measure such that, for G V × ∈ U ⊗ V almost every z, i.e., U˜0 is of full measure. ￿ each w (z, s) E, = ∈ It is now easy to prove the main result of this section. T lim s G N (z) b(w). N = ˇ →∞ Theorem 3.9. Let G be a polarization field and U have finite S G measure. Then, for every θ, Uθ is of full measure in Uθ , i.e., the n ˜ Proof of Proposition 3.7. Clearly − (w) W(0, 2πn w) N sequence G (θ, z) converges for µd -almost every z Uθ to ϕ(0, 2πn z) P = ; = S ∈ ; and thus (θ, ) N (θ, ) Ψ(0, 2πn z)s ˆG z as N . Moreover, every component of G ; S 1 →∞ S · ￿ ￿ is in (Uθ , θ ,µd ) and converges to the corresponding ( n(w)) ( ( , π ) )T (ϕ( , π )) L U 1 b − Ψ 0 2 n z s G 0 2 n z component of G (θ, ) in and each component of G (θ, ) P = T T ; ; 1 Sˆ · L Sˆ · s Ψ (0, 2πn z)G(ϕ(0, 2πn z)) is in (Uθ , θ ,µd ). = [ ; ; ] L U sT Ψ (2πn, 0 ϕ(0, 2πn z)) G(ϕ(0, 2πn z)) , G = [ ; ; ; ] Proof of Theorem 3.9. Because of Lemma 3.6, U˜0 is of full measure in U. Since ϕ(0,θ ) is measure preserving, (3.24) where the third equality follows from (3.13). Therefore by g g ;· T 1 N 1 n gives µd (Uθ ) µd (U0 ) µd (U). (3.31) s G N (z) N n −0 b( − (w)) and, because b is in = = = = P N (θ, ) 1(U V, ,µ µ ), Proposition 3.7 follows from Since G is a bounded function and U is of finite d 3 S · N 1 L × U ⊗ V ￿× measure, every component of (θ, ) is in (Uθ , θ ,µ ). Theorem 2.4. ￿ SG · L U d Thus, by Definition 3.4, and as a consequence of the dominated ( , ) convergence theorem, every component of N (θ, ) converges To go from the bˇ of Proposition 3.7 to the ˆG 0 of G S · 1 S · Lemma 3.6 we need a technical lemma. in to the corresponding component of G (θ, ). In particular, L 1 Sˆ · each component of G (θ, ) is in (Uθ , θ ,µd ). ￿ Proposition 3.8. Let F be of full measure and define Sˆ · L U ∈ U ⊗ V the section of F at s V by Fs z U (z, s) F . Then Remarks. ∈ := { ∈ : ∈ } the following hold. (10) In the trivial case where 0 (hence Ψ I3 3) one has A = = × (a) There exists a VF V of full measure such that, for s VF , N 1 s ⊂ ∈s N 1 − µd (F ) µd (U), i.e., for µ3-almost every s V, F is of (0, z) G(ϕ(0, 2πn z)), (3.36) = ∈ SG = ; full measure in U. N n 0 ￿= (b) There exist three linearly independent vectors s1, s2, s3 in so that in this case the proof of Lemma 3.6 is VF . straightforward—one just applies Theorem 2.4 to the 142 J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149

components of G! However if 0 then Ψ I3 3 at each (θ, z), so that Ploc 1. These things are discussed for A ￿= ￿= × | |≤ so that one cannot apply Theorem 2.4 to the components example in [30]. of G because (3.36) does not hold. Thus if 0 an The orbital density is a normalized LD as defined in A ￿= alternative has to be found. Our above approach is to Section 2.2.1. In order to apply Theorem 3.9, we assume that U apply Theorem 2.4 to the function b in (3.34) and then has finite measure. We also assume that the beam is in “orbital use Proposition 3.8(b) to prove Lemma 3.6. Accordingly, equilibrium”, i.e. that the orbital density ρg is 2π-periodic in the proof of Lemma 2.5 is considerably simpler than that measure. The polarization P(θ) of the whole beam, i.e., the of Lemma 3.6. beam polarization, is defined as (11) With Remark (10) we see that ρ and behave very g SG differently under stroboscopic averaging, i.e. the ergodic P(θ) ρ (θ, z)P (θ, z)dµ (z). (3.37) := g loc d theorem has very different meanings for the orbit and ￿Uθ spin motion. Thus it is not surprising that the convergence Recalling from Remark (8) that ρg Ploc is a polarization field, speeds of the Cesaro` sequences are different. See also the integral in (3.37) is well defined because ρg(θ, )Ploc(θ, ) Section 3.4.2. · · is bounded and U has finite measure. Since ρg Ploc is a (12) It is clear by Remark (7) and by Theorem 3.5 how one polarization field it has a generator G, i.e., ρg Ploc G where constructs an invariant spin field if G has no zeros. = S Sˆ G(z) ρg(0, z)Ploc(0, z). One can show more generally that an invariant spin field = If the beam is in “spin equilibrium”, i.e. if Ploc is 2π-periodic exists if G (0, ) is nonzero µd -almost everywhere. On Sˆ · in measure, then clearly G is a polarization field 2π-periodic the other hand we do not know if such a always S SˆG in measure and the beam polarization is 2π-periodic by (3.37). exists. Nevertheless our experience suggests that invariant If the beam is not in spin equilibrium, one considers the spin fields always exist. However, as we know from [4, time average of the beam polarization. Then, although the local Section 6], there are examples where no C1 invariant spin polarization is not 2π-periodic in measure, the time average field exists. of the beam polarization exists and is 2π-periodic as we now (13) If a positive constant ξ exists such that, for all N and z, show. The time average at some θ, usually corresponding to the ( ( , ))T N ( , ) ξ ( , ) G 0 z G 0 z then, trivially, ˆG 0 has no S SG ≥ S · position of a particle physics experiment in the ring, is zeros on U˜0 . This criterion plays an important role in N 1 the computation of invariant spin fields with the computer − code SPRINT [32,33,42] and these computations show P(θ) lim (1/N) P(θ 2πn). (3.38) ¯ := N + →∞ n 0 that the criterion is very often fulfilled. ￿= (14) The simplest example for f with U of finite measure is Using the theory we have developed, we first show that P¯ exists. the betatron motion of Appendix A. However, since f We conclude from (3.37) that is much more general, our spin–orbit system can cover N 1 situations where the orbital motion is nonintegrable. The 1 − P(θ 2πn) typical nonintegrable examples we have in mind for f + N n 0 with U of finite measure are orbital motions which obey ￿= N 1 the Moser twist theorem and where U is bounded. 1 − ￿ ρ (θ 2πn, z)P (θ 2πn, z)dµ (z) = g + loc + d N n 0 Uθ 2πn 3.4. Several aspects of polarization fields including applica- ￿= ￿ + N 1 tions to the physics of polarized beams 1 − G (θ 2πn, z)dµd (z) = N Uθ 2πn S + n 0 ￿ + In this section we outline various aspects of polarization ￿= fields and their Cesaro` sequences and in particular we elaborate N (θ, z)dµ (z). (3.39) = SG d on our comments in Section 1 on the significance of our ￿Uθ work for polarized beam physics in storage rings. See also the Clearly the integrals in (3.39) are well defined. By reference list and Section I in [4]. The basics of polarized beam N 1 Theorem 3.9, the components of G (θ, ) converge in to physics are discussed in [27, Section 2.7], [33]. S · L the corresponding components of G (θ, ) so that by (3.39) Sˆ · N 1 (see also [11, Theorem 15.1]) limN (1/N) n −0 P(θ 3.4.1. Long time averages of polarization N →∞ = + 2πn) limN (θ, z)dµd (z) G (θ, z)dµd (z). = →∞ SG = Sˆ ￿ The statistical properties of a beam of spin-1/2 particles Thus P exists. ¯ ￿ ￿ (e.g. protons, electrons, positrons, muons) can be encoded π It is easy to show that P¯ is 2 -periodic once one has in a one-particle quantum mechanical density matrix [24]. A established that P is bounded. The latter holds since particle beam in a storage ring represents a highly mixed quantum state. The orbital density ρ ρg and Ploc are then P(θ) ρ (θ, z) P (θ, z) dµ (z) = | |≤ g | loc | d obtained by a Wigner–Weyl transform in the semiclassical ￿Uθ approximation [5,6]. The local polarization is a polarization ρ (θ, z)dµ (z) 1, field and it is the local average of the normalized vector s/ s ≤ g d = | | ￿Uθ J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 143 where we used the fact that P 1 and that the LD ρ is Stroboscopic averaging was originally motivated by the | loc|≤ g normalized. need to calculate invariant spin fields since, as explained in For particle physics experiments it is important to maximize Section 1 and in Section I in [4], invariant spin fields are key P(θ) as a function of P . This is discussed in [4, Section I], objects for systematizing spin motion in storage rings. See | ¯ | loc [33]. also Section 3.4.3 below. An invariant spin field is derived from a 2π-periodic polarization field obtained by stroboscopic 3.4.2. The stroboscopic averaging algorithm and SPRINT averaging by simply normalizing, as outlined in Remark We now describe a simple and transparent spin–orbit (12), assuming that the stroboscopic average has no zeros. tracking algorithm for calculating the stroboscopic average of Stroboscopic averaging provides a general and powerful means polarization fields. to calculate invariant spin fields as well as to inquire about Consider a polarization field i.e., fix the generator G. If their existence. Hence it represents a major addition to the SG z U G and if N is sufficiently large then N (θ, z) is a good tool kit for describing polarized beams in storage rings. For ∈ ˜θ SG approximation of (θ, z). Now consider N (0, z0) where we example, it can be applied to nonintegrable orbital motion SˆG SG hold z0 U G fixed. Then and for integrable orbital motion it even works on orbital ∈ ˜0 resonance. See also Section 3.4.5. For references to algorithms N (0, z0) for computing invariant spin fields see [27]. SG N 1 Remark 1 − . Ψ(2πn, 0 ϕ(0, 2πn z0))G(ϕ(0, 2πn z0)) = ; ; ; (15) We now briefly comment on the rate of convergence. A N n 0 ￿= high convergence speed for a Cesaro` sequence is of course N 1 1 − crucial to the success of the tracking algorithm outlined (Ψ( 2πn, 0 z0))TG(ϕ( 2πn, 0 z0)), (3.40) = − ; − ; in this section and since SPRINT has been in operation N n 0 ￿= much experience has been accumulated. One finds that if where we used (3.20) in the first equality and (2.6), (3.12) the orbital motion is integrable the convergence is usually and (3.13) in the second. With (3.40) we see that in order to linear in N in the sense that for a given z0 there exists a compute N (0, z0) we only need to compute G at the set of SG real constant c such that for all N ϕ( π , 0) N 1 points 2 n 0 z n −0 and multiply by the matrices in N 0 N f 0 { − ; 0} =T N 1 G (0, z ) G (0, z ) c the set (Ψ( 2πn, 0 z )) n −0 . This clearly reduces the task S S , (3.43) { − ; } = N 0 N f N 0 ￿ G (0, z ) − (0, z0) ￿ ≤ N of computing G (0, z ) to the backtracking of the trajectory, ￿|S | G ￿ S ￿ |S |￿ defined by the initial value problem ￿ N 0 N f ￿ 0 where￿ G (0, z ) and G (￿0, z ) are assumed to be S N S z f (θ, z), z(0) z0, nonzero and f (0, z0) is assumed to be approximately ˙ = = G 0 S D Ψ(θ, 0 z0) (θ, z)Ψ(θ, 0 z0), (3.41) G (0, z ). Experience for high energy storage rings shows 1 ; = A ; Sˆ 0 that a few thousand turns usually suffice, i.e. N f Ψ(0, 0 z ) I3 3, = ; = × 10 000 is an appropriate order of magnitude. The linear over N 1 turns backward around the storage ring. Moreover, it convergence law (3.43) was derived analytically, by using − is simple to calculate N (0, z) for z ϕ(2πn, 0 z0) by further plausibility assumptions, in [32]. This is a subject for SG = ; forward or backward tracking using, for sufficiently large N, further investigation. the relations Note that for LD’s there seems to exist no linear law analogous to (3.43). For the convergence speed of N (0,ϕ(2πn, 0 z0)) N (2πn,ϕ(2πn, 0 z0)) SG ; ≈ SG ; the Cesaro` sequences of LD’s, see for example [40, Ψ(2πn, 0 z0) N (0, z0), (3.42) Section 3.2]. Recall also Remark (11). ￿ = ; SG where in the equality we used the fact that N is a polarization 3.4.3. The uniqueness of invariant spin fields SG field and where in the approximation we used (3.25) and that While, as pointed out in Remark (8), there may be a large ϕ(2πn, 0 z0) U G . To summarize, by tracking a single number of 2π-periodic polarization fields, this does not imply ; ∈ ˜0 trajectory of the system (3.41) and with (3.40), (3.42), we can that there is a large number of 2π-periodic spin fields and we compute N (0, z) for all points z of the form z ϕ(2πk, 0 z0) address this issue here. SG = ; where k is an integer. For integrable orbital motion, we can approach the Of course, one then repeats this procedure with a sufficiently uniqueness of the stroboscopic average of a polarization field dense grid of z0’s until a sufficiently dense portrait of the by appealing to existing results on the uniqueness of the N function G (0, ) has been obtained. Then one can use (3.22) invariant spin field. For integrable orbital motion the domains S · N to compute the functions (θ, ) for a sufficiently dense grid Uθ decompose into tori and for every torus a “spin tune” can SG · of θ’s in the interval 0, 2π . These are the key ideas in [32] and be defined which determines if the system is on “spin–orbit [ ] they led to the stroboscopic averaging algorithm in SPRINT [8, resonance” (the spin tune, as a function of the torus, is the 33,42]. In SPRINT, G is usually taken to be a constant function “amplitude dependent spin tune” mentioned in Section 1). Note so that the calculations are further simplified. that the ellipses Eε,θ in Appendix A are the tori for the betatron 144 J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 motion. In particular, if for a given torus the system is not on and such that, if z0, z1 M and z0 z1, then M(z0) M(z1) ∈ ˜ ￿= ∩ = orbital resonance and not on spin–orbit resonance, we know . Thus for every z U we have a unique integer n(z) and ∅ ∈ that a 2π-periodic spin field on that torus is unique up to a a unique point z0(z) M such that z ϕ(2πn(z), 0 z0(z)) ∈ ˜ = ; sign if it is of class C1 [4]. One expects quite generally that all and such that, if z M, z0(z) z. Of course, for every ∈ ˜ = invariant spin fields on such a nonresonant torus are essentially z M, we have n(z) 0 and it easily follows from (2.5) ∈ ˜ = identical in a measure-theoretic sense, i.e. modulo µ -nullsets. and (2.6) that for every z U and every integer k we have d ∈ It thus follows from Remark (12) that in the above situation z0(ϕ(2πk, 0 z)) z0(z) and n(ϕ(2πk, 0 z)) n(z) k.We ; = ; = + all stroboscopic averages, which have no zero, point into the now use the axiom of choice again to choose a spin vector s(z) essentially unique direction given by the invariant spin field for every z M and we define a function G U R3 by ∈ ˜ : → on the given torus. Calculations with SPRINT have provided G(z) Ψ(2πn(z), 0 z0(z))s(z0(z)). (3.45) a large amount of numerical evidence for this behavior. While := ; we do not dwell further on the issue of uniqueness in the present Using (3.11) and (3.12) and the above mentioned properties of paper, let us mention that the procedure would rest upon Fourier z0(z) and n(z) it follows that G, defined by (3.45), satisfies analysis in the orbital angle as one can see for the C1 case (3.17), whence (θ, z) defined by (3.14) is 2π-periodic in θ. SG in [4]. The only additional ingredient in the present context, Of course if s does not vanish everywhere then neither does where we don’t have invariant spin fields which are C1 in the G. Moreover G is normalized if s is. Since the function s orbital angle, is Parseval’s equality [31]. is basically arbitrary, we have shown that a large supply of The uniqueness in the case of nonintegrable orbital motion normalized functions G U R3 exists which satisfy (3.17). : → is more complicated to deal with and to our knowledge has not It is clear that if a 2π-periodic polarization field exists, then SG been seriously studied. Of course, since SPRINT is a tracking the function s can be chosen such that (3.45) holds, since, if algorithm, it can be used to look closely at this issue. z M, then G(z) s(z). However (3.45) alone gives no ∈ ˜ = hints for choosing s in a way such that the G in (3.45) is - U 3-measurable, i.e., fulfills the regularity conditions. Thus the 3.4.4. The “filling-up” method R filling-up method does not solve the problem of the existence We now comment on the consequences of relaxing the of 2π-periodic spin fields and it is also easy to see that it also regularity conditions of Section 3.2.1. does not solve the problem of existence of invariant spin fields. It is trivial that polarization fields exist in great numbers Note that the problem of the existence of normalized 2π- since every bounded and measurable G gives a polarization periodic LD’s is, at least in the case of a U of finite measure, field . In other words, the generator G has only to fulfill SG considerably simpler. In fact if U is of finite measure then, the regularity conditions. However the existence of invariant there exists a constant positive g, such that ρg is a 2π-periodic spin fields is, except for trivial spin–orbit systems, a nontrivial normalized LD. matter. For example, Theorem 3.5 does not guarantee that has no zeros. If is a polarization field 2π-periodic in 3.4.5. The map formalism SˆG SG measure then, as we know from Section 3.2.1, G is not only Although in this paper we have chosen to work with flows, bounded and measurable but also has to fulfill the dynamical we will now comment on the use of one-turn maps. condition (3.17) almost everywhere. Conversely, we also know In this paper we have restricted our discussion to functions 1 from Section 3.2.1 that is 2π-periodic in measure if G fso of class C in order to be consistent with the situation SG satisfies (3.17) almost everywhere. As far as we are aware, in real storage rings and the treatment in [4]. But in practice, it is, by analytical means, nontrivial to discover whether a G simulations are often made with the approximation that the exists which solves (3.17) almost everywhere and which is magnets have hard edged magnetic fields or are represented by 1 nonzero almost everywhere, given our regularity conditions on thin lenses so that the requirement that fso(θ, w) be of class C G. However, with stroboscopic averaging one has a numerical with respect to θ must be replaced by the weaker requirement means to investigate the matter of existence. If we relax the that it is only piecewise C1 in θ. It is then usually convenient regularity conditions it is known that the dynamical condition to analyze the spin motion using one-turn maps instead of (3.17) allows a large class of normalized solutions. This can be differential equations and to study the spin–orbit motion on the θ 0 section, having chosen the time origin so that f (0, ) is shown with the so-called “filling-up” method [34]. Although = so · C1. The dynamics is then determined by and the definitions this method does not solve the above problem of existence of P an appropriate G, it does give further insight into the nature of of LD and polarization field are modified accordingly. For invariant spin fields. Thus we will now explain it. example, the 2π-periodicity in measure becomes a one-turn For brevity we only treat the case of nonperiodic orbital periodicity in measure and the definition of the 2π-periodic motion in the sense that, for every z0 U and every polarization field is essentially replaced by (3.17). Thus [4] and ∈ nonzero integer n, ϕ(2πn, 0 z0) z0. Defining M(z0) this paper can be easily reformulated in terms of maps. One 0 ; ￿= := thereby obtains essentially the same theory as with the flow n ϕ(2πn, 0 z ) , we know, by the axiom of choice, that ∈Z{ ; } formalism. a subset M of U exists such that 1 ￿ ˜ An interesting example where fso is only piecewise C U M(z0), (3.44) is provided by the following important model. In this model = z0 M orbital motion is integrable and has just one degree of freedom ￿∈ ˜ J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 145 and the ring includes two thin lens Siberian Snakes which setting to prepare the reader for its use in Section 3. Section 2.4 thus provide two discontinuities in the θ dependence of the gives a summary and Appendix A applies the theory to an in (3.2). This model is often called the “single resonance important example from beam dynamics. A model with two Siberian Snakes” [9,10]. Outside the snakes, Section 3.4 covers several aspects of polarization fields corresponds to the model discussed in Section VII in [4]. including applications to the physics of polarized beams. In A For the single resonance model with two Siberian Snakes one Section 3.4.1 we applied Theorem 3.9 to the polarization of can write a simple analytical expression for . Thus analysis the whole beam. Section 3.4.3 introduces the reader to the P is especially simple and spin–orbit tracking simulations can be important issue of the uniqueness of invariant spin fields and made to run very efficiently. As always, when is piecewise Section 3.4.4 discusses a side issue. The final Section 3.4.5 A C1 in θ, is C1. outlines the important extension of our work from the flow P We expect for systems where is C1 that an invariant spin formalism of [4] to the map formalism. P field exists which is C1. See the models in [4]. However, there With this paper we have established that the simulation are exceptional models where this is not true. For example when code SPRINT, outlined in Section 3.4.2, computes periodic the single resonance model with two Siberian Snakes is on polarization fields, i.e., functions which, in particular, obey orbital resonance at a so-called “snake resonance tune” and for regularity conditions. This fact was not clear when SPRINT was most values of the remaining parameters of the model, invariant first created. While numerical work with SPRINT has involved spin fields exist but none of them is C1. See Fig. 12 in [10] and integrable and bounded orbital motion, our work here does not the accompanying discussion. require integrability. In addition, SPRINT has been applied to discontinuous functions f and . Thus it would be interesting A 4. Summary and discussion to extend SPRINT to nonintegrable orbital motion and to extend our approach to discontinuous functions f and . In Section 3 of this paper, we consider an initial assignment A of spins, G, which evolves according to the flow W Acknowledgments of f by (θ, z) Ψ (θ,0 ϕ(0,θ z)) G (ϕ(0,θ z)). so SG = ; ; ; We call a polarization field. For each G, we have SG We wish to thank V. Balandin, D.P. Barber, G.H. constructed a 2π-periodic polarization field G (θ, z) N N Sˆ N =1 Hoffstaetter, and M. Vogt for useful discussions. We are also limN 1U G (z) G (θ, z) where G (θ, z) (1/N) n −0 →∞ ˜θ S S = = indebted to the late Gerhard Ripken and to D.P. Barber for G (θ 2πn, z) and U Uθ is of full measure when encouragement and a careful reading of the manuscript in its SG + ˜θ ⊂ ￿ µ (U)< , as we proved using the ergodic theorem. This various stages. d ∞ is the main result of our paper. The flow W is defined after (3.3) and the polarization Appendix A. Appendix on the example of betatron motion field in Definition 3.1. The stroboscopic average ˆG of G S G S We will now illustrate the machinery of Section 2 with an is defined in Definition 3.4 and the convergence set U˜θ of N (θ, ) is defined in (3.23). The main results of this section important model of linearized particle motion in a storage ring. SG · The main result of this section is the explicit construction of the are Theorems 3.5 and 3.9. In Theorem 3.5 we prove that SˆG stroboscopic average in Proposition A.1. is a 2π-periodic polarization field but obtain no information on U G the measure of ˜θ . In Theorem 3.9 we use the ergodic theorem A.1. Basic properties G to show that U˜θ is of full measure in Uθ . As in the orbital case the stroboscopic average may be zero almost everywhere. As explained in Section 1, a storage ring has a closed orbit An important problem left unsolved in this paper is that which is a periodic curve in R3 defined by dipole magnets. A of the conditions under which the stroboscopic average of a storage ring is designed, using magnets, so that the motion of polarization field has no zeros. This problem is closely related particles starting near the closed orbit is stable. In Frenet–Serret with the problem of existence of the invariant spin field since coordinates defined on the closed orbit the transverse linearized (see Remark (12)) a stroboscopic average which has almost no motion is given by a pair of Hill equations (thus d 4). = zeros implies the existence of an invariant spin field. Another Restricting to one degree of freedom one has d 2 and one = problem which we did not investigate is the long time stability Hill equation, of 2π-periodic polarization fields. T In the proof of Lemma 3.6, which is central to the proof z f (θ, z) (z2, k(θ)z1) , (A.1) of Theorem 3.9, the reader can see how we have overcome ˙ = := − π 1 the main technical obstacle in this paper which is the fact (see where k is a 2 -periodic C function determined by the also Remark (10)) that the ergodic theorem cannot be applied quadrupole magnets which serve to focus the beam. “directly” to polarization fields because they have three, and Eq. (A.1) is a special case of the more general problem not just one, components. In contrast the proof of Lemma 2.5 which we will treat in this section, namely applies the ergodic theorem “directly” since Liouville densities T z f (θ, z) D1 H(z,θ) S(θ)z, (A.2) are scalar quantities. ˙ = := J = J Section 2 laid the framework for the orbital part of Section 3 defined by the Hamiltonian H(z,θ) 1 zT S(θ)z (see = 2 and it also discussed the ergodic theorem and its use in a simple also [38]). Here S is a 2π-periodic, symmetric, 2 2 matrix × 146 J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149

1 01 We are now in a position to derive the most important property of class C , 10, and S is sometimes called J := − J of the Hamiltonian system (A.2), namely a conservation law. It a Hamiltonian matrix￿ [26].￿ In the case of (A.1), S(θ) = diag(k(θ), 1). is easy to see that the PSM satisfies The principal solution matrix (PSM) Φ is defined by ΦT(θ,θ )C(θ)Φ(θ,θ ) C(θ ), where 0 0 = 0 D Φ(θ,θ ) S(θ)Φ(θ,θ ), Φ(θ ,θ ) I, (A.3) γ(θ)α(θ) 1 0 = J 0 0 0 = C(θ) T J(θ) , (A.10) := J = α(θ)β(θ) and is symplectic, i.e., ΦT Φ , due to the symmetry of S. ￿ ￿ J = J The general solution (2.3) can be written as by simply showing that the derivative of the lhs with respect to 0 θ is zero. Since solutions of (A.2), with z(θ0) z , are given 0 0 = ϕ(θ,θ0 z ) Φ(θ,θ0)z . (A.4) by z(θ) Φ(θ,θ )z0, it follows that ; = = 0 The f given by (A.2) fulfills the conditions imposed in T 0 T 0 z C(θ)z (z ) C(θ0)z . (A.11) Section 2.1. In particular, U R2 is an open nonempty set = = satisfying (2.8). We will obtain a more appropriate U shortly. In particular The period advance matrix for the θ0 section is given by T 2 2 e(z,θ) z C(θ)z γ(θ)z1 2α(θ)z1z2 β(θ)z2 (A.12) P(θ0) Φ(θ0 2π,θ0). (A.5) := = + + := + is a constant of the motion, i.e., e(z,θ) is constant along Due to (2.6) we have Φ(θ 2π,θ 2π) Φ(θ,θ ), so that P + 0 + = 0 solutions of (A.2). is 2π-periodic as it must be. Also, we note that P is symplectic Since C(θ) is symmetric and positive definite, then for each and that it satisfies θ, the set of points z R2 for which ∈ P(θ) S(θ)P(θ) P(θ) S(θ). (A.6) 2 2 ˙ ε e(z,θ) γ(θ)z 2α(θ)z1z2 β(θ)z , = J − J = = 1 + + 2 Since the trace of the rhs of (A.6) is zero, Tr P is independent (0 ε< ) (A.13) [ ] ≤ ∞ of θ. We assume that S is such that the solutions of (A.2) are forms an ellipse which we denote by E . Since e is a constant stable, i.e., the elliptic case where Tr P < 2. Thus there are ε,θ | [ ]| of the motion, we have two values of ν (0, 2π)such that Tr P 2 cos ν. In addition ∈ [ ]= P (θ) is nonzero, for if it were zero then P (θ)P (θ) would 12 11 22 ϕ(θ,θ0 Eε,θ0 ) Eε,θ . (A.14) be 1 and we would have the contradiction that Tr P ; = | [ ]| = P (θ) 1/P (θ) 2. Since sin ν is nonzero and unique Because Eε,θ 2π Eε,θ , we obtain | 11 + 11 |≥ + = up to a sign we fix it by choosing it to have the same sign as ϕ(θ0 2πn,θ0 Eε,θ ) Eε,θ . (A.15) P (θ). In summary, there is a unique value of ν (0, 2π) + ; 0 = 0 12 ∈ such that Tr P 2 cos ν and P (θ) sin ν>0. In particular, the motion under (A.2) in each θ0 section is [ ]= 12 Remark. confined to a single ellipse. It follows that 2 (16) Note that P(θ) is a 2π-periodic solution of (A.6). We do U Eε ,0 z R e(z, 0)<ε (A.16) := ￿ ={ ∈ : } not claim that all solutions of (A.6) are 2π-periodic nor is 0 ε <ε ≤￿￿ this required. ￿ is, for every ε>0, a nonempty open invariant set Given ν we now construct, for every θ, an important satisfying (2.8), i.e., Φ(2π, 0)U U. Furthermore by (A.14), = representation of the symplectic matrix P(θ). Since sin ν 0 Uθ Φ(θ, 0)U ϕ(θ, 0 U) ϕ(θ, 0 Eε , ) 0 ε￿<ε ￿ 0 ￿= = = 2 ; = ≤ ; = we can define functions α and β by P11(θ) cos ν α(θ) sin ν Eε ,θ z e(z,θ)<ε . =: + 0 ε￿<ε ￿ R and P (θ) β(θ) sin ν. Noting that Tr P 2 cos ν and that ≤ ={ ∈ : ￿} 12 =: [ ]= Finally, we construct the PSM Φ. Consider the symplectic det P 1, we obtain the representation ￿ = matrix transformation P(θ) I cos ν J(θ) sin ν, ζ T (θ)z, T T T (A.17) = + = J = J α(θ)β(θ) J(θ) , (A.7) which gives := γ(θ) α(θ) ￿− − ￿ ζ ( (θ) (θ))ζ, 1 where γ(θ) (1 α2(θ))/β(θ). It is clear that β(θ),γ(θ) > 0 ˙ R Sˆ where R TT˙ − and := + = J + := −J (A.18) and that the functions α,β,γ are uniquely determined by P T 1. Sˆ T − ST− and are 2π-periodic and C1. Of course P determines ν and it := is interesting to note that it has two distinct and θ-independent It follows from the symplecticity of T that R and Sˆ are ν ν symmetric whence (R S) is a Hamiltonian matrix. Our aim eigenvalues: cos i sin . From (A.6) and (A.7) J satisfies J + ˆ ± is to find T so that (A.18) can be integrated and we proceed (θ) (θ), T J˙ S J J S (A.8) by looking for T such that ζ ζ is a constant of the motion, = J − J T T T 2 i.e., ζ ζ z T Tz is constant. From (A.11), this will be true and, since J I , T = 2 2 =− if T T C. This gives T T γ,T11T12 T21T22 = 11 + 21 = + = P exp(ν J). (A.9) α, T 2 T 2 β and from symplecticity, T T T T 1. = 12 + 22 = 11 22 − 12 21 = J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 147

We find that these can be solved by taking T 0 and we A.2. Computing the stroboscopic average of an arbitrary LD 12 = obtain From Theorem 2.6 the stroboscopic average of every LD is a 1 β− 2 0 2π-periodic LD. Here, we construct an explicit formula for ρg T 1 1 . (A.19) ˆ = ￿αβ − 2 β 2 ￿ where U in (A.16) is the domain of g. Note that U depends on ε>0. From the definitions in (A.18) a short calculation using (A.8) We begin by constructing a candidate for ρ . From (2.18) we ˆg gives R S (S22/β)I and clearly the solution of (A.18) for have +0ˆ = ζ(θ0) ζ is = N 1 θ N 1 − 0 S22(θ ￿) ρg (θ, z) g (Φ(0,θ 2πn)z) . (A.24) ζ exp(ψ(θ,θ0) )ζ ,ψ(θ,θ0) dθ ￿. = N + = J = β(θ ) n 0 ￿θ0 ￿ ￿= (A.20) Note that for every z Uθ there is a unique ε in 0,ε) such ∈ ￿ [ 1 1 0 that z belongs to Eε ,θ whence by (A.14) the sequence of points Since z T − (θ)ζ T − (θ) exp(ψ(θ,θ0) )T (θ0)z the ￿ = = J Φ(0,θ 2πn)z belongs to the ellipse Eε , . Thus for fixed (θ, z) PSM for (A.2) is + ￿ 0 the sequence (A.24) samples points on the ellipse Eε￿,0 and so (θ,θ ) 1(θ) (ψ(θ,θ ) ) (θ ). ρg(θ, z) can depend only on values of g on that ellipse. That Φ 0 T − exp 0 T 0 (A.21) ˆ = J this is the case will be seen explicitly in (A.28) and (A.29). Remark. Because the stroboscopic average ρg is an LD, (2.24) holds ρ ( ˆ, ) (17) We could have proceeded by looking for T with T12 0 and so we only need to determine g 0 . By (A.9) and (A.22) = n ˆ · 1 such that R S ωI for scalar ω, and the result would we have that Φ(0, 2πn) P− (0) T − (0) exp( nν )T (0) + ˆ = = = − J be the same. ￿ and so by (A.24) (θ ) (ν (θ )) N 1 It follows from (A.21) that P 0 exp J 0 where N 1 − 2π = ˆ ρg (0, z) g (exp( nν J(0))z) ν S22/βdθ and we have used the fact that = N − ˆ = 0 n 0 ￿= 1 ￿ T − T J. (A.22) N 1 1 − 1 J = g(T − (0) exp( nν )T (0)z). (A.25) = − J Thus N n 0 ￿= 2π S (θ) If z U, then z lies on an ellipse Eε ,0 and, by (A.15), ν 22 dθ mod 2π. (A.23) ∈ ￿ = β(θ) the points exp( nν J(0))z lie on that ellipse. Accordingly the ￿0 − points ζn exp( nν )T (0)z lie on the circle with radius Since the transformation (A.17) is measure preserving and = − J ε￿ centered at the origin and if ν/2π is rational the average since the set TUθ is the interior of the circle with radius is easily computed as Proposition A.1(b) will show. If ν/2π √ε µ (U ) πε centered at the origin, we have 2 θ , which is is irrational then the ζ lie densely on the circle and we can θ = n independent of . apply the Weyl equidistribution theorem [36, Section 3] if g is Remark. continuous. It also follows, by the continuity of the function (18) In the case of (A.1), we have derived the Courant–Snyder e( , 0), that if ν/2π is irrational and z Eε ,0, x R then all · ∈ ￿ ∈ description of betatron motion (see also [27, Sec- points exp( xJ(0))z lie on Eε , (in particular if z U then − ￿ 0 ∈ tion 2.1.1]) which was developed for describing particle exp( xJ(0))z U). We conclude that if ν/2π is irrational and − ∈ motion in circular accelerators in the 1950s. In this case, e g is continuous then ρ N (0, z) converges for all z U and g ∈ is called the Courant–Snyder invariant to honor the impor- N tant contribution of Courant and Snyder in their classic pa- ρg(0, z) lim ρg (0, z) g(z), (A.26) ˆ = N =¯ per [15]. An interesting discussion of various approaches →∞ to (A.11) can be found in [16]. We have followed the spirit where the function g U 0, ) is defined by 2π ¯ : →[∞ of [38] in our study of (A.2). g(z) (1/2π) g(exp( xJ(0))z)dx and where we have ¯ := 0 − The calculation of α and β is central to the design of used (A.22). From (2.24) and (A.26) ￿ storage rings and algorithms for their construction can be 1 2π found in standard accelerator physics texts [21]. From the ρg(θ, z) g (exp( xJ(0))Φ(0,θ)z) dx, (A.27) ˆ = 2π − approach above it is clear that J can be determined by ￿0 calculating P as defined in (A.5). Another approach would and this is our candidate for the stroboscopic average. In fact be to solve (A.8) with an appropriate initial condition the above is a proof that it is, in the case where g is continuous. 2 subject to γ(θ0) (1 α (θ0))/β(θ0). In this context Note that since exp( xJ(0))z U if ν/2π is irrational and := + − ∈ it would be interesting to understand the set of all z U, it is clear that the integrals in the definition of g and ∈ ¯ solutions to (A.8) and in particular the subset given by in (A.27) are well defined if ν/2π is irrational. This may seem 2 γ(θ0) (1 α (θ0))/β(θ0). This reduces to the study of ν := + odd but recall from (A.7) that J and are intimately related. Φ(θ, 0)ZΦ(0,θ)which for an arbitrary constant matrix Z Note that in this appendix we abbreviate dµ1(x) and dµ2(x) by is the general solution of (A.8). ￿ dx. 148 J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149

If g is not continuous we cannot apply the equidistribution and where in the second equality of (A.33) we used the theorem but we nevertheless know from Lemma 2.5 that transformation theorem for Lebesgue integrals. To prove (A.31) N ρg (0, z) converges for µ2-almost every z in U and so we expect we need two lemmas which we prove after completing the proof for irrational ν/2π that (A.26) holds for µ2-almost every z in of Proposition A.1. U. The following proposition, which is a consequence of the Lemma A.2. If h 1(U, ,µ ) is a continuous function, ergodic theorem, confirms this expectation. ∈ L U 2 A and x 0, 2π then ∈ I ∈[ ] Proposition A.1. Let f be given by (A.2) in the elliptic case, i.e., Tr P < 2 and let U z R2 e(z, 0)<ε where h(z)dz h(z)dz. (A.34) | [ ]| := { ∈ : } exp( xJ(0))A = A ε>0. Then for every LD ρg the following hold. ￿ − ￿ (a) If ν/2π is irrational then, for µ2-almost every z Uθ , Lemma A.3. For all x in 0, 2π and A ∈ [ ] ∈ I 1 2π ρg(θ, z) g (exp( xJ(0))Φ(0,θ)z) dx, (A.28) f (x) g(z)dz. (A.35) ˆ = 2π − = ￿0 ￿A where Φ is the PSM and J is given by (A.7). 1 2π ν/ π ν/ π / > We conclude from (A.35) that f (x)dx g(z)dz (b) If 2 is rational, i.e., 2 q p with integers p 0 2π 0 = A = whence, by (A.32), it follows that (A.31) holds. This completes and q, then ￿ ￿ the proof that g is a version of E(g ). p 1 ¯ |I 1 − Now, since g is a version of E(g ) and since, by ρg(θ, z) g (exp( nν J(0))Φ(0,θ)z) . (A.29) ¯ |I ˆ = p − Lemma 2.5, ρg(0, ) is also a version of E(g ), we conclude n 0 ˆ · |I ￿= that (A.26) holds for µ1-almost every z in U. Thus by Theorem 2.6, ρ is an LD so that (A.28) holds for µ -almost Remark. That U is nonempty and open, and satisfies (2.8) was ˆg 2 every z Uθ . shown in Appendix A.1. Without loss of generality we take ∈ ￿ ε 1/π so that µ2 is a probability measure. Proof of Proposition A.1(b). Let ν/2π be rational, i.e., ν/2π = q/p with integers p > 0 and q. We showed earlier in this Proof of Proposition A.1(a). The proof follows easily once = section that exp( nν J(0))z U if z U whence the sum in we prove that the function g, defined after (A.26), is a version − ∈ ∈ ¯ (A.29) is well defined. Let of E(g ), where A ϕ(2π, 0 A) A A |I I := { ∈ U : ; = }={ ∈ 1 P(0)A A . We also showed earlier in this section that gn g (exp( nν J(0))z) g(T − (0) exp( nν )T (0)z), U : = } := − = − J the integral in (A.28) is well defined. then gn p gn and We first show that g is - -measurable. Since g is bounded + = ¯ I R and - -measurable we easily find that g is - -measurable. p 1 lp 1 N 1 N 1 − − − U R ¯ U R ρ (0, z) gn Thus the proof is complete if we show that g is invariant. g = N +···+ + 1 ¯ ￿n 0 n (l 1)p n lp￿ Now exp( xJ(0)) T − (0) exp( x )T (0) and P(0) ￿= =￿− ￿= 1 − = − J = T − (0) exp(ν )T (0) whence p 1 N 1 J 1 − 1 − 2π l gn gn, (A.36) 1 1 = N + N g(P(0)z) g(T − (0) exp((ν x) )T (0)z)dx n 0 n lp ¯ = 2π − J ￿= ￿= ￿0 2π where l is the integer part of (N 1)/p. Clearly the second term 1 1 − g(T − (0) exp( x )T (0)z)dx on the rhs goes to zero as N and limN (l/N) 1/p = π − J →∞p 1 →∞ = 2 0 whence ρ (0, z) (1/p) − g(exp( nν J(0))z). Since ρ ￿ ˆg = n 0 − ˆg g(z), (A.30) is an LD, we conclude that (A.29)= holds. ￿ =¯ ￿ where in the second equality we used the fact that exp( x ) is We now prove the two lemmas. − J 2π-periodic in x. Proof of Lemma A.2. Clearly Secondly we need to show that for A ∈ I h (exp( xJ(0))z) dz h(z)dz. (A.37) g(z)dz g(z)dz. (A.31) A − = exp( xJ(0))A A ¯ = A ￿ ￿ − ￿ ￿ Since h is continuous, it follows that h (exp( xJ(0))z) is Integrating g over A and using the Fubini theorem we have − ¯ continuous in x whence by (A.37) and a lemma on parameter 1 2π dependent integrals (see e.g. [11, Lemma 16.1]) the lhs of g(z)dz f (x)dx, (A.32) (A.34) is continuous in x. A ¯ = 2π 0 ￿ ￿ For every integer n we have by (A.9) and (A.22) where f 0, 2π R is defined by :[ ]→ A Pn(0)A exp(nν J(0))A = = f (x) g (exp( xJ(0))z) dz 1 := − T − (0) exp(nν )T (0)A. (A.38) ￿A = J Since ν/2π is irrational it follows from (A.38) that for a dense g(z)dz, (A.33) = exp( xJ(0))A set of x-values in 0, 2π we have exp( xJ(0))A A. It ￿ − [ ] − = J.A. Ellison, K. Heinemann / Physica D 234 (2007) 131–149 149

follows that for a dense set of x-values in 0, 2π , (A.34) [10] D.P. Barber, M. Vogt, New J. Phys. 8 (2006) 296. [ ] holds. Since the lhs of (A.34) is continuous in x, the claim [11] H. Bauer, Measure and Integration Theory, Walter de Gruyter, Berlin, follows. 2001. ￿ [12] P. Billingsley, Probability and Measure, 3rd ed., Wiley, New York, 1995. [13] L. Breiman, Probability, Society for Industrial and Applied Mathematics, Proof of Lemma A.3. Let g1, g2,... be a sequence of con- tinuous functions in 1(U, ,µ ) converging in 1 to g as Philadelphia, 1992. L U 2 L [14] C. Chicone, Ordinary Differential Equations with Applications, Springer, n , i.e., g (z) g(z) dz 0 as n (such a →∞ U | n − | → →∞ New York, 1991. sequence exists due to [11, Theorem 29.14]). By Lemma A.2 [15] E.D. Courant, H.S. Snyder, Ann. Phys. 3 (1958) 1–48. ￿ we have for all x 0, 2π [16] H. Qin, R.C. Davidson, Phys. Rev. ST Accel. Beams 9 (2006) 054001. ∈[ ] [17] Ya.S. Derbenev, A.M. Kondratenko, Sov. Phys. 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