A Dynamical Approach to Validated Numerics

A dynamical approach to validated numerics O. Jenkinson and M. Pollicott In memorium, A.B. Katok ∗ Abstract We describe a method, using periodic points and determinants, for giving alternative expressions for dynamical quantities (including Lyapunov exponents and Hausdorff dimension of invariant sets) associated to analytic hyperbolic systems. This leads to validated numerical estimates on their values, 1 Introduction 1.1 An overview There are many well known and important numerical invariants in the context of dynamical systems, including for example entropy, Lyapunov exponents, and the Hausdorff dimension of invariant sets. Even in the context of uniformly hyperbolic systems, many of these invariants do not have simple explicit expressions, nor are they easy to estimate. This leads naturally to the search for useful alternative expressions for each of these quantities which, in particular, lend themselves to accurate numerical evaluation. Our aim in this account is to describe an approach developed over some years, based on periodic points for (hyperbolic) dynamical systems. In particular, in addition to presenting a general overview we will also give a general recipe for converting this information into formulae and useful numerical estimates for a variety of characteristic values, such as those mentioned above. Perhaps the most important aspect of this side of the work is that we obtain rigorous bounds on the errors. Given the data available, the idea is to minimise the error estimate, subject to the practical constraints imposed by the computational power available. There is always some flexibility in the choices, which we can exploit in order to obtain the best error estimate, i.e., the smallest bound on the error. Typically, the quantities that we can expect to study in this approach are those that can be expressed in terms of the thermodynamic pressure. The main ingredient in our approach is a complex function (called the determinant) which packages together the data on periodic orbits and from which can be derived estimates on the pressure, and consequently the quantities of interest. In our presentation, no specialist knowledge is required to implement the general result. How- ever, for completeness we give a fairly complete outline of the proof (with some details on operator theory postponed to the appendix). ∗The second author collaborated with A. Katok, G. Knieper and H. Weiss over 30 years ago on the regularity of topological entropy for Anosov flows. One of the methods used there was based on the characterisation of the entropy as a pole for the dynamical zeta function, which is a complex function closely related to the determinant central to ideas we pursue here. 1 1.2 An historical perspective 1 INTRODUCTION 1.2 An historical perspective The starting point for our approach is the important work of Grothendieck in the 1950s on nu- clear operators [17, 18]. This extended the classical theory of trace class operators and Fredholm determinants [12]. Although the impact of this theory in functional analysis and operator theory was well understood, it was not until 20 years later that Ruelle employed it with great effect in ergodic theory and dynamical systems, in his application to dynamical zeta functions [40]. This viewpoint was highly influential in the work of mathematical physicists (e.g., the well known study by Cvitanovićin his monumental online tome [7]). With the advent of modern computers, an important component in ergodic theory and dynamical systems has been the focus on explicit computation of quantities arising in the context of dynamical systems. Typically, these approaches are based on the quantity in question being expressed in terms of an associated Ruelle transfer operator, implicitly assumed to act on some appropriate space of functions, and then a finite dimensional approximation to the operator is used to reduce this to a finite dimensional matrix problem, to be solved numerically. In contrast, the approach we will describe is to exploit real-analytic properties of the underlying dynamical system by introducing Ruelle transfer operators with strong spectral properties (in particular nuclearity) which allows us to exploit the earlier circle of ideas initiated by Grothendieck. 1.3 A selection of applications By way of an appetiser to this approach, we now list a cross section of actual and potential applications to a number of different areas in mathematics. We will elaborate these later, but for the purposes of motivation we list them here. 1. In number theory, a recent breakthrough in the understanding of the Markov and Lagrange spectra M; L ⊂ (0; +1) from Diophantine approximation has been brought about by the work of Matheus & Moreira who were able to estimate the dimension of the difference M n L of these spectra, obtaining first a lower bound [34] and then an upper bound [35]. The methods, in both cases, involved the approximation of the dimension of certain fractal sets which would be amenable to the techniques developed in [25] for proving rigorous high quality bounds on the dimension (cf. Example 4.9 (b)). 2. In the field of spectral geometry, there is a strong tradition of computing eigenvalues of the Laplacian, dating back to pioneering work of Hejhal [20] using classical methods. However, McMullen's approximations for the lowest eigenvalue for certain infinite volume hyperbolic manifolds were based on the dimension of the limit set and using this viewpoint could be accurately estimated [32]. 3. Within dynamical systems, among the most widely studied numerical quantities are Lyapunov exponents, measuring the exponential instability of solutions to various problems. There is also interest in estimating Lyapunov exponents in the theory of random matrix products; in particular, in information theory this leads to the computation of entropy rates for binary symmetric channels, related to examples of hidden Markov chains [19]. We will develop this last setting in the following subsection. 1.4 Illustrative Example: Lyapunov exponents for Bernoulli interval maps We begin by specifying a suitable class of hyperbolic transformations. In particular, we want to assume that the systems we are studying are both real analytic and uniformly hyperbolic, for 2 1.4 Illustrative Example: Lyapunov exponents for Bernoulli interval maps 1 INTRODUCTION 1 1 0 2 1 Figure 1: The graph of an expanding map of R=Z represented on the unit interval. example either a real analytic expanding map, or a real analytic hyperbolic diffeomophism or flow on a locally hyperbolic set. Before formulating statements in greater generality, let us consider a very specific example of ! a one dimensional map. Let X = R=Z be the unit circle and let T : X ! X be a piecewise C Bernoulli map of the interval which is expanding, i.e., there exists λ > 1 such that jT 0(x)j ≥ λ for all x. In this setting it is well known that there is a unique T -invariant probability measure µ dµ 1 which is absolutely continuous with respect to Lebesgue measure (i.e., dx 2 L (X)) by the famous Lasota-Yorke theorem (see [28, x5.1]). Example 1.1. More concretely, suppose T : R=Z ! R=Z is defined by T (x) = 2x + sin(2πx) 1 0 3 (mod 1) where jj < 4π , so in particular jT (x)j > (2 − 2π) > 2 > 1 for all x 2 R=Z. The Lyapunov exponent of the unique absolutely continuous T -invariant probability measure µ is defined by Z L(µ) = log jT 0(x))j dµ(x) ; (1) and by the well known Rohlin identity this also equals the entropy h(µ). We briefly summarise the method for estimating L(µ) in three steps. Step 1 (Complex functions and coefficients). We wish to consider period-n points T nx = x and then define for t 2 R the coefficients a1(t); a2(t);::: using the Taylor expansion 1 ! 1 X zn X j(T n)0(x)j−t X exp − = 1 + a (t)zn + ··· : n 1 − ((T n)0(x))−1 n n=1 T nx=x n=1 Step 2 (Coefficients and Lyapunov exponents). It can be shown that the Lyapunov exponent L(µ) given by (1) admits the alternative formulation P1 0 − n=1 an(0) L(µ) = P1 ; n=1 nan(0) 3 2 HYPERBOLIC MAPS where both the numerator and denominator are absolutely convergent series. Truncating these series to give the computable quantity − PN a0 (0) L = n=1 n ; N PN n=1 nan(0) we note that LN ! L as N ! +1 ; and so for a large natural number N, the quantity LN is an approximation to L. Step 3 (Error bounds). The quality of the approximation to L given by LN may be indicated heuristically by comparing how closely LN and LN−1 agree, though more accurate errors in the approximation can be obtained using: 1. A certain value θ 2 (0; 1), which we refer to as the contraction ratio, measuring the extent to which a complex disc D is mapped inside itself by the inverse branches Tj of T ; 2 2. The integrals β = 1 R 1 P T 0(x + re2πit)(T (x + re2πit))k dt, where 1 ≤ k ≤ L; and k r2k 0 j j j q PL 2 3. The weights αk = l=k+1 βl for N ≤ k ≤ L, where N < L are suitably chosen. In particular, we can then use these values to bound the coefficients an(t), with n > N, for which it is impractical to explicitly compute them with effective error estimates. This will be explained in greater detail in x5. Now that we have illustrated the general theme using the specific example of Lyapunov exponents for one dimensional expanding maps, we can turn to more general dynamical settings, and more general characteristic values.

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