Fractal Curvatures and Minkowski Content of Self-Conformal Sets

FRACTAL CURVATURES AND MINKOWSKI CONTENT OF SELF-CONFORMAL SETS

TILMAN JOHANNES BOHL

ABSTRACT. For self-similar fractals, the Minkowski content and fractal curvature have been introduced as a suitable limit of the geometric characteristics of its parallel sets, i.e., of uniformly thin coatings of the fractal. For some self-conformal sets, the surface and Minkowski contents are known to exist. Conformal iterated function systems are more flexible models than similarities. This work unifies and extends such results to d general self-conformal sets in R . We prove the rescaled volume, surface area, and curvature of parallel sets converge in a Cesaro average sense of the limit. Fractal Lipschitz- Killing curvature-direction measures localize these limits to pairs consisting of a base point in the fractal and any of its normal directions. There is an integral formula. We assume only the popular Open Set Condition for the first-order geometry, and remove numerous geometric assumptions. For curvatures, we also assume regularity of the Euclidean distance function to the fractal if the ambient dimension exceeds three, and a uniform integrability condition to bound the curvature on “overlap sets”. A limited converse shows the integrability condition is sharp. We discuss simpler, sufficient conditions. The main tools are from ergodic theory. Of independent interest is a multiplicative ergodic theorem: The distortion, how much an iterate of the conformal iterated function system deviates from its linearization, converges.

1. INTRODUCTION

d Let F R be a self-conformal set, i.e., invariant under ﬁnitely⊆ many contractions φi whose differen-

tials φi0 (x) are multiples of orthogonal matrices, [ F = φi x : x F . i I ∈ ∈ Any self-similar set F is a simple, “rigid” special case. We assume the well-known Open Set Condition to keep overlaps φi F φj F small. Classical volume∩ and curvature are not meaning- arXiv:1211.3421v1 [math.MG] 14 Nov 2012 fully applicable to the geometry of F . So we approximate F with thin “coatings”, its parallel sets FIGURE 1.1.

Fr : x : min x y r , r > 0. Dark: self-conformal set F , IFS = y F ∈ − ≤ of three Möbius maps (6.0.1). The limits of the rescaled Lebesgue measure and Shaded: a parallel set Fr . Lipschitz-Killing curvature-direction measures of Fr

2000 Mathematics Subject Classiﬁcation. primary: 28A80, 28A75, 37A99; secondary: 28A78, 53C65, 37A30. Key words and phrases. self-similar set, self-conformal set, curvature, Lipschitz-Killing curvature- direction measure, Minkowski content, conformal iterated function system. Supported by grant DFG ZA 242/5-1. The author has previously worked under the name Tilman Johannes Rothe. 1 CURVATURES OF SELF-CONFORMAL SETS 2 do describe F geometrically. The purpose of this work is to make that statement precise under minimal assumptions. The type of limit is most easily explained using the example of the well-known Min- kowski content. We localize it as a measure version. Denote by Cd (Fr ,A) the Lebesgue measure of F A (if A d ). The rescaling factor r D d keeps r D d C F , from vanish- r R − − d ( r ) ing trivially as∩r 0. (Thermodynamic⊆ formalism determines for the correct· dimension D > 0, see below.)→ But the rescaled total volume can oscillate. Cesaro averaging on a log- arithmic scale suppresses this. Without any further assumptions, we prove the localized average Minkowski content converges (as a weak limit of measures):

1 Z frac 1 D d d r Cd (F, ) := lim r − Cd (Fr , ) · ε 0 lnε · r → | | ε T Z 1 t D d ( ) t = lim e − − Cd (Fe , )d t . T T − · →∞ 0 Curvature and area ﬁt into this approach as well as volume. Classically, they are in- trinsically determined and form a complete system of certain Euclidean invariants. To d illustrate classical curvature, let K R be a compact body whose boundary ∂ K is a smooth submanifold. Write n (x) for⊆ the unit outer normal to K at x ∂ K . The differential n x of the Gauss map n has real eigenvalues x ,..., ∈ x (geometri- 0 ( ) κ1 ( ) κd 1 ( ) cally, inverse radii of osculating spheres).1 They are called principal curvatures− of K at x. The full-dimensional Hausdorff measure d 1 coincides with the Riemannian vol- − ume form on ∂ K . Lipschitz-Killing curvature-directionH measures are integrals of symmetric polynomials of principal curvatures: For integers 0 k < d and any Borel set A d Sd 1, deﬁne ≤ R − ⊆ × Z X d 1 Ck (K ,A) := const 1 (x,n (x)) A κj1 (x) κjd k 1 (x) d − (x). − − { ∈ } j1<

Our parallel sets Fr will only satisfy a positive reach condition rather than smoothness. (Parallel sets do not depend on the choice of φi and enjoy widespread use in singular curvature theory.) Their curvature-direction measures are deﬁned via geometric measure theory. Each point x can have a different cone of normals. Anisotropic curvature quantities also prove useful to describe heterogeneous materials, see [STMK+11] and the references therein. Therefore, we prefer curvature-direction measures rather than d non-directional Federer curvature, i.e., than their R marginal measures. We prove the average fractal curvature measures

1 Z frac 1 D k d r Ck (F, ) := lim r − Ck (Fr , ) · ε 0 lnε · r → | | ε converge under mild assumptions. While most literature uses this normalization, a different one displays more clearly frac Ck (F, ) does not depend on the ambient dimension (Remark 3.6 below). ·

1Curvature is second-order geometry because n x is a second-order coordinate derivative, as opposed 0 ( ) to Minkowski content and various Hausdorff and packing measures. CURVATURES OF SELF-CONFORMAL SETS 3

We ﬁx some notation for the assumptions. This dynamical system powers all convergence results: The left shift on F inverts the generating contractions, i.e., σ φi x := x. Thermodynamic formalism deﬁnes both a unique dimension D > 0 and conformal probability measure µ on F . The contraction ratio is its potential, µ φωm φω1 F = R D φ φ d µ, ω I . But the curvature is covariant under◦ the ··· ◦ similarity F ωm ω1 0 k ◦ ··· ◦ ∈ φi0 x rather than σ. Also, only a similarity lets us compare its image of Fr locally with any thinner parallel set. (This restricts us to conformal φi .) The distortion ψ captures the mismatch between iterations of the dynamical system and their differentials (Figure 2.1). For an I -valued sequence ω and points x,y F , we prove convergence ∈ 1 ψ y : φ φ x − φ φ y φ φ x , ωßm,x = ωm ω1 0 ωm ω1 ωm ω1 | ◦ ··· ◦ ◦ ··· ◦ − ◦ ··· ◦ ψω,x y : lim ψ y. e = m ωßm,x →∞ | frac Two mild conditions guarantee Ck (F, ) converges (see Theorem 3.3). Neither condition applies to the volume (k = d ), or to· the surface area measure (k = d 1) if restricted to d (instead of d Sd 1). − R R − (1) Positive reach of the× closed complement of almost every parallel set both of

F and of almost every distorted fractal ψω,x F (Assumption 3.1). This guaran- tees existence and continuity of curvature measures. It is automatically satis- fied in ambient dimensions d 3. In higher dimensions, it is more convenient to check that almost all values≤ of the Euclidean distance function are regular. var (2) A uniform integrability condition limits the Ck mass on overlaps of images under different φi (Assumption 3.2 below). This is sharp due to a limited converse (Remark 4.4). Let B (x,r ) be an r -ball around x. A sufficient, integrability condition is that M 1 1 X r,ω,x r k sup − C var ψ F ,ψ B x,ar , a > 1 ( ) − k ωßm,x ωßm,x ( ) M >0 M r 7→ m=0 | | belongs to a Zygmund space related to µ. The much stronger, sufficient condition k var esssupx F,r >0 r − Ck (Fr , B (x,ar )) < ∈ ∞ intuitively means the rescaled principal curvatures should be uniformly bounded. We are the first to prove average fractal curvature exists for general self-conformal sets because our Ck contains principal curvatures when k < d 1. Our theorem encom- passes all previous convergence results about (D-dimensional)− average Minkowski or surface content of self-conformal sets [KK10, FK11, Kom11]. We remove all previously needed geometrical conditions on the Open-Set-Condition set or on the hole struc- ture. Fractal curvatures or their total mass were first investigated for self-similar sets [Win08, Zäh11, RZ10, WZ12, BZ11]. The Minkowski content was treated earlier [LP93, Fal95, Gat00], and in a very general context recently in [RW10] and [RW12]. frac There is an integral formula for the limit Ck (F, ) (Theorem 3.3). It is always finite. The formula shows the Minkowski content is strictly· positive. But for k < d , the inter- frac pretation as fractal curvature should be checked if Ck (F, ) vanishes trivially [Win08]. · d Unsurprisingly, the fractal curvature is just as self-conformal as F . Its R marginal measure must therefore be a multiple of the conformal probability µ. That means the fractal locally looks the same almost everywhere. Its Sd 1 sphere marginal measure in- − fib herits a group action invariance from the self-conformality (f ω (2.1.10) is invariant under Gηu (2.1.9) below). Otherwise, it is free to provide geometric information beyond CURVATURES OF SELF-CONFORMAL SETS 4 its total mass. For example, the Sierpinski gasket resembles an equilateral triangle, and both have the same sphere marginals up to a constant [BZ11]. (But the group spreads it uniformly over Sd 1 for generic , I . Future research could ask how curvature ap- − φi frac proaches its limit [LPW11].) In the self-similar case, Ck (F, ) is an (independent) product of both its marginal measures. In the self-conformal case,· it is a skew product. But the skewing factor is easy to compute and only takes values in the orthogonal group of orth d (the factor ψ y in (2.1.10) below). R η0e,u ˆ The motivation for this work is threefold. Fractal curvature is a geometric parame- ter to distinguish between sets of identical dimensions. Statistical estimates of various fractal dimensions already play a role in the applications. It is natural to expect further geometric parameters to be useful, both in their own right and to improve estimators for dimensions. Secondly, curvature (geometry) determines analytic properties. In the classical, smooth Riemannian case, the Dirichlet Laplacian eigenvalue counting function has an expan- sion in terms of the boundary’s curvatures. Heat kernel estimates also involve curvature. In one dimension, the next-to-leading order spectral asymptotics are proven: Let A be an open set whose boundary ∂ A has a nontrivial, D-dimensional Minkowski content without Cesaro averaging (modified Weyl-Berry conjecture), as λ : → ∞ 1/2 frac 1 D/2 D/2 # eigenvalues λ = constC1 (A)λ + constDC1 ∂ A,A S λ +o λ . ≤ × Upper and lower bounds hold at the λD/2 order if the upper and lower Minkowski content are nontrivial [LP93]. In several dimensions, true curvature exists, and the conjecture fails. We hope to provide some missing geometric input. Thirdly, there is the long-standing quest in geometric measure theory to extend the notion of curvature as far as possible. Classes of such “classical” sets include C 2-smooth manifolds, convex sets, sets of positive reach, and their locally finite unions [Fed59, Zäh86, RZ01]. See [Ber03a] and the references therein for an overview. Support measures generalize to all closed sets but loose additivity and other characteristic features of curvature [HLW04]. They and our fractal curvature usually live on mutually disjoint subsets of F , e.g. for the Sierpinski gasket. The central ideas of the proof are explained in the respective section headings. The methods are dynamical although the results are geometric. We reduce the conformal case to similarities by separating into the distortion and its derivative , which φ ψ φ0 matches Euclidean covariance. The normal directions necessitate the two-sided shift orth dynamical system extended by the orthogonal group co-cycle . φ0

2. PRELIMINARIES 2.1. Self-conformal sets and dynamics. Conformal maps are automatically (conjugate) 2 d holomorphic in R or even Möbius in R , d > 2 (Liouville theorem, e.g. [Geh92, Theo- rem 1.5]). Assumption 2.1. Throughout this entire paper, φi : i I , 2 I < will be a conformal iterated function system (IFS), i.e., each function ∈ ≤ | | ∞

φi : V V → 1 γ d is an injective C + -diffeomorphism deﬁned on an open, connected set V R such that the differentials x are linear similarities of d , and ⊆ φ0 ( ) R

φi x φi y smax x y , x,y V − ≤ − ∈ CURVATURES OF SELF-CONFORMAL SETS 5 for some global smax < 1. We assume the Open Set Condition (OSC): There is an open, connected, bounded, nonempty set X◦ such that X V, ⊆ φi X◦ X◦ , i I ⊆ ∈ and φi X◦ φj X◦ = ∅, i = j , i , j I . ∩ 6 ∈ Without loss of generality ([PRSS01]), the set X shall satisfy the Strong Open Set Con- dition X◦ F = ∅. 1 γ∩ 6 The IFS shall extend conformaly in C + to the closure of an open set VMU V : φi : VMU VMU . ⊇ → This is the situation studied in [MU96], except we allow only ﬁnitely many maps φi . Their cone condition is not needed for ﬁnite I , and their bounded distortion condition can be proved [MU03]. See [MU96] for further discussion and facts given without a reference. S The self-conformal fractal is the unique, invariant, compact set ∅ = F = i I φi F X. The (Borel probability) conformal measure µ on F and dimension D6 > 0 are uniquely∈ ⊆ characterized by Z Z X D (2.1.1) f (x) d µ(x) = φi0 x f φi x d µ(x). i I F F ∈

n Abbreviate x n := x1 ...xn I , the reversed word xgn := xn ...x1, φx n := φx1 S n | φx , I : | I . We will∈ identify µ-almost all points| x F with their (unique)◦ ··· ◦ n ∗ = n N0 coding sequence∈ x x I N, i.e., x lim v for any∈ starting v V . The 1 2 − n φx n ( ) (left) shift σ : F F ···or ∈σ : I N I N ≡maps x→∞to φ| 1x. The unique, equivalent,∈ σ- F F x−1 invariant probability→ ν is ergodic.→ Denote its Hölder continuous density p := d ν/d µ. The proofs need the two-sided shift space I N F . Let σbi (ω1ω2ω3 ..., x1x2x3 ...) := × (x1ω1ω2 ..., x2x3x4 ...) for (ω,x) I N F ; perhaps it is most natural to think of ω in ∈ × reverse order: σbi ((...ω2ω1)(x1x2 ...)) = ((...ω1x1)(x2x3 ...)). Denote µbi, νbi, pbi = d νbi/d µbi the Rokhlin extensions. We extend it again, by the orthogonal group, to I N F O (d ). The new measure × × νbi := νbi O(d ) is the direct product with the Haar probability on O (d ). The new left shift is ⊗ H

orth 1 m m m − (2.1.2) σbi ω,x, g := xßm.ω, σ x, φx0 m σ x g , | | orth 1 i.e., the skew product with the Rokhlin co-cycle (ω,x) φx0 1σx − . Recall some basic properties. The Perron-Frobenius7→ operator| gives ([PU10, Lemma 4.2.5]): Z Z g g (2.1.3) x d ν x φ x d ν ω,x , g L µ. ( ) ( ) = ωÞn bi ( ) 1 p p | ∈ F I N F × ¦ © Denote B (x,r ) := y : y x r . F is a D-set for µ (Ahlfors regular): there is a 0 < C F < , − ≤ ∞ CURVATURES OF SELF-CONFORMAL SETS 6

1 µB (x,r ) (2.1.4) C F− D C F . ≤ r ≤ Bounded distortion is fundamental. There is a global constant 0 < K < such that for all x,y V , I , ∞ MU τ ∗ ∈ ∈ x 1 φτ0 (2.1.5) K − K . y ≤ φτ0 ≤ There is a constant 0 K such that for all x V , r x dist x,V c , I , < φ < MU / φτ0 MU τ ∗ [MU96, (BDP.3)]: ∞ ∈ ≤ ∈ ! 1 r (2.1.6) B φτx,r Kφ− φτ B x, B φτx,r Kφ . x ⊆ φτ0 ⊆

The distortion, how much the non-linear map φω n deviates from its differential (up to isometries), converges exponentially (Proposition| 4.8 below): 1 ψ y : φ x − φ y φ x ,(2.1.7) ωÞn,x = ωÞ0 n ωÞn ωÞn | | | − | ψω,x y : lim ψ y.(2.1.8) e = n ωÞn,x →∞ | Note its derivative is point-wise a similarity. The Brin ergodicity group is the compact subgroup of O (d ) generated by

® orth 1 orth ¸ − orth (2.1.9) Gηu : ψ x φ x ψ φi x : x F,i I . = η0e,u i0 η0e,u ∈ ∈ O(d ) Let be its Haar probability. For f : V Sd 1 , the average of f p over the ergodic G − R / ﬁbreHthrough (ω,x) is × → Z Z orth 1 orth ﬁb − (2.1.10) f ω (z ,n) := f yˆ, ψη0 ,u yˆ gˆ ψω0 ,u z n d G gˆ d µ yˆ . e e H F Gηu

N fib Different choices of η,u I F do not alter f ω (but do conjugate Gηu ). So Gηu and fib f can be skipped when replacing∈ × u with x. Readers not interested in directional Ck ω R should ignore all underbars and set G : O d , f fib z ,n : f d µ. ηu = ( ) ω ( ) = F 2.2. Curvature. We will approximate F with its r -parallel set (Minkowski sausage, dila- tion, offset, homogenization) Fr or the closure of the complement: ¦ d © c Fr := x R : x y r for a y F , Fer := (Fr ) .(2.2.1) ∈ − ≤ ∈ d The reach of a closed set K R is the supremum of all s > 0 such that every point x Ks ⊆ x y ∈ has a unique nearest neighbor y K . If reach K > 0, the set of pairs y, forms the x−y ∈ Federer normal bundle, nor K Fed59 . Positive reach replaces the C smoothness| − | often [ ] ∞ assumed in differential geometry. We will implicitly extend any conformal or similarity map of d to one of d Sd 1, ξ R R − orth z ,n : z , z n (i.e., to the cotangent bundle equipped with the× Sasaki ξ( ) = ξ( ) ξ0 ( ) metric). Here, orth is the orthogonal group component of its derivative . ξ0 ξ0 We will need only these properties of Lipschitz-Killing curvature-direction measures d Ck : Let K R be a set of positive reach. For k 0,...,d 1 , Ck (K , ) is a signed Borel measure on⊆ nor K d Sd 1 with (locally) finite∈ { variation.− } It is Euclidean· motion R − ⊆ × CURVATURES OF SELF-CONFORMAL SETS 7 covariant and homogeneous of degree k , i.e., if g is a similarity with ratio s > 0 and orthogonal group part g orth, then for Borel A d , N Sd 1, R − ⊆ ⊆ orth k (2.2.2) Ck g K , g A g N = s Ck (K ,A N ). × × It is locally determined, i.e., if two sets of positive reach K , Kˆ agree K G = Kˆ G on an open set G d , then for A d , N Sd 1, ∩ ∩ R R − ⊆ ⊆ ⊆ (2.2.3) Ck (K ,(A G ) N ) = Ck Kˆ,(A G ) N . ∩ × ∩ × d A special case of continuity is stated in Fact 4.15 below. The R projection of Cd 1 (K , ) agrees with half the d 1 -dimensional Hausdorff measure d 1 (surface area)−on the· ( ) − boundary ∂ K . Because− the Lebesgue measure d and the rotationH invariant probability d d 1 d 1 on the sphere share these properties, weL define for Borel A , N S , S R − H − ⊆ ⊆ d Cd (K ,A N ) := (K A) Sd 1 (N ). × L ∩ H − d d 1 (The trivial product with d 1 makes every C live on S .) We define the curva- S k R − H − × ture measure of Fr via the normal reflection since we assume 0 < reach Fer (Assumption 3.1 below): for Borel A d , N Sd 1, k d , R − < ⊆ ⊆ d 1 k (2.2.4) Ck (Fr ,A N ) := ( 1) − − Ck Fer ,A n : n N . × − × {− ∈ } This is consistent in case both Fr and Fer are (unions of) sets of positive reach ([RZ01, Theorems 3.3, 3.2]). Of course, Ck (Fr , ) enjoys the above geometrical properties. All assertions in this paper also hold for (non-directional)· Federer curvature measures, i.e., their projection C F , Sd 1 onto d . (We conjecture projecting to d does not re- k r − R R duce the variation’s mass· ×[BZ11, Rem. 3.15].) For a gentle introduction to curvature, combine the brief summary in [Zäh11] with [SW92] (polyconvex case), [KP08] (currents and Hausdorff measure), [Fed59] (positive reach), [Zäh86] (curvature-direction measure), [Ber03a] (survey of notions). To put curvature in context: The total masses K C K , d Sd 1 form a complete k R − system of Euclidean invariants in the following sense.7→ Every set-additive,× continuous, motion invariant functional on the space of convex bodies is a linear combination of total curvatures (Hadwiger’s theorem). By means of an approximation argument, this holds for large classes of singular sets, including sets of positive reach [Zäh90]. Further- more, Ck is the integral of symmetric functions of generalized principal curvatures κi over the d 1-dimensional Hausdorff measure d 1 on the normal bundle: − − P H Z κi x,n ...κi x,n i 1< < εmax − ( ) Let η,u I N F be any fixed reference point. Define for x,z V , r >≤0, ∈ × ∈ x z (2.3.1) A ( x z ,r ) := max 1 | − | , 0 | − | − ra a “smoothed-out” indicator function of the ar -ball around x. Inspired by R (x,m) S ∼ dist x,∂ τ I m φτX , the renewal epochs are defined (Definition 4.10) for µ-almost all ∈ CURVATURES OF SELF-CONFORMAL SETS 8

FIGURE 2.1. Left: fractal F (black), parallel set Fr (both shades), point x = (1,1,2,3,1,...) (marked) surrounded by a ball of 2a times its renewal radius R (x,1). The ball supports A and localizes curvature, so it must avoid φ2F r φ2F r (light gray). ∪ Middle: level set φ1F r containing x. Right: Magniﬁcation is the composition of the dynamical system map σ and distortion by ψ. Black/shaded: magniﬁed φ1F r or ψ1,x F . The marked point ψ1,x σx is surrounded with r / φ10 x 2aR (σx,0| ). |

x F and m N as: ∈ ∈ ( 1 ) c 2K aε dist(x,X ) φ− max Re (x,0) := min 1, c , 2Kφa maxx F dist(x,X ) ∈ R x,m : sup n x R n x,0 .(2.3.2) ( ) = φx0 n σ e (σ ) n m | ≥ Denote the measure theoretic entropy of ν and σ,

Z (2.3.3) Hν : D ln φ σx d ν x . = x0 1 ( ) − F

Define for x F , r,s,t 0, , f : V Sd 1 Borel measurable, with the shorthand ( εmax] − R z z ,n ∈d Sd 1, the∈ measure × → = ( ) R − ∈ × Z 1 r r D k A x z ,r k , (s,t ] ( ) − ( ) (2.3.4) ρ s,±t x,r, f := R | − | f z d Ck± Fr ,z , ( ] A y z ,r d µy norfFr − and for a finite word I in (2.1.7) and an infinite word I N in (2.1.8): τ ∗ ω ∈ ∈ k , (2.3.5) πτ ± x,r, f := r D k 1(R(x,1),R(x,0)] r A ψτ,u x ψτ,u z ,r Z ψ x − 0τ,u − f z d C ψτ,u F ,ψτ,u z , R D k± r ψ y A ψ y ψ z ,r d µy 0τ,u τ,u τ,u − CURVATURES OF SELF-CONFORMAL SETS 9

(2.3.6) πk x,r, f : ωe = r D k 1(R(x,1),R(x,0)] r A ψω,u x ψω,u z ,r Z ψ x − e e 0ωe,u − f z d Ck ψω,u F ,ψω,u z R D e r e ψ y A ψ y ψ z ,r d µy ω0 ,u ωe,u ωe,u e − if these integrals exist, and ρk : ρk ,+ ρk , , πk : πk ,+ πk , , s,t = (s,t ] (s,−t ] τ = τ τ − ( ] − − ρk ,var : ρk ,+ ρk , , πk ,var : πk ,+ πk , . (s,t ] = (s,t ] + (s,−t ] τ = τ + τ −

The above transform into each other, see (4.5.4). The proofs deﬁne further symbols: Kφ (2.1.6), cψ Kψ Proposition 4.8, CA (4.4.3), q (4.5.3).

3. MAIN RESULT AND LIMITFORMULA S Recall F = i I φi F is self-conformal with (OSC), see Assumption 2.1. ∈ Assumption 3.1. (Regularity of parallel sets) Throughout this entire paper, whenever k < d , we will assume the following. Lebesgue almost all r > 0 shall be regular values of the Euclidean distance function to F and to F for -almost every . ψωe,u µbi ω Alternatively, let almost all r > 0, ω I N satisfy ∈ (1) reach Fer > 0, (2) if y,m nor Fer , then y, m / nor Fer , ∈ − ∈ (3) reach F 0. ψåωe,u r > If the surface area measure C is restricted to d instead of d Sd 1, the assumption d 1 R R − is needed only for k < d 1. − × − This is always satisﬁed in ambient dimensions d 3 due to [Fu85]. It is also true for any strictly self-similar set whose convex hull is a polytope≤ ([Pok11]). Regular distance values imply the alternative conditions, see [Fu85, Theorem 4.1], [RZ03, Proposition 3]. Assumption 3.2. (Uniform integrability) Assume at least one of the following: (1) k = d (Minkowski content), (2) k = d 1 (Surface area content), (3) − k var ◦ esssupr >0,x F r − Ck Fr , B (x,ar ) < , ∈ ∞ µ-essential in x, Lebesgue in r . (4) (Zygmund space) M 1 1 X− h r,ω,x : r k sup C var ψ F ,ψ B◦ x,ar ( ) = − k ωßm,u ωßm,u ( ) M >0 M r m=0 | | satisﬁes both

R(x,0) Z Z d r max 0,h (r,ω,x) lnh (r,ω,x) d νbi (ω,x) < , { } r ∞ I N F R(x,1) × ε max Z Z var ◦ 1(max R(x,0),ε ,εmax](r )Ck Fr ,B(x,ar ) d r { } sup0<ε εmax k d νbi ω,x < . 2 ln(εmax/ε)r ( ) ≤ r ∞ I N F 0 × CURVATURES OF SELF-CONFORMAL SETS 10

(5) As above, except integrating h lnh from R (x,1)/K to R (x,0) K and with h (r,ω,x) :=   M 1 1 X−  r  r k sup C var ψ F ,ψ B◦ x,ar 1 . − k ωßm,u ωßm,u ( ) (R(x,1),R(x,0)]   M >0 M r   m=0 | | ψ x ω0ßm,u | (6) The family of functions (3.0.1)   1 X § ª ,x,r k ,var x,r,1 1 r u k ,var x,r,1 (ω ) εmax ρ max R x,0 , , ( ) + φ > ε π ( ) ln ( ( ) ε εmax] ω0ßm ωßm 7→ ε { } m N | | ∈ is uniformly r 1d r d ,x -integrable for all parameters 0, 2 . − νbi (ω ) ε ( εmax/ ) (7) As above, except the integration variable x replaces u , including∈ inside πk ,var. ωßm | Intuitively, the assumption limits how much curvature the overlap sets may carry. More precisely, let , I be different shortest words such that the diameters of τ ϑ ∗ ψ φ F and ψ φ ∈F are smaller than ra 1/ 2. Then to check (uniform) integra- ωßm,u τ ωßm,u ϑ − p | k ,var | k var ◦ bility of its π - or r Ck ψ F , -mass, only the intersection of ψ B (x,ar ) ωßm − ωßm,u r ωßm,u | | · | with the union of overlap sets ψ φτF ψ φϑ F of all such pairs τ, ϑ ωßm,u r ωßm,u r | ∩ | matters. The curvature on the non-overlap portion of ψ B◦ x,ar satisfies item (3) ωßm,u ( ) above, see [Zäh11, Theorem 4.1]. | In case of self-similar fractals, further sufficient conditions are the Strong Curvature Bound Condition [Win11] and polyconvex parallel sets [Win08, Lemma 5.3.2]. Their advantage is they do not involve parallel sets widths r of all orders. (In our Zygmund space condition, x R (x,1) is bounded from below if the Strong Separation Condition holds.) 7→ Not all self-conformal sets satisfy item (3) (see [RZ10]), hence the more general conditions. fib Recall π (2.3.6) and f ω (2.1.10). Theorem 3.3. Let k 0,...,d , and F be a self-conformal set (Assumption 2.1). Suppose Assumptions 3.1 (only∈ { if k d } 1) and 3.2 hold. Then for any continuous f : V Sd 1 − ≤ − × → R, the following limit exists and is finite: ε Zmax frac 1 D k d r (3.0.2) Ck F, f := lim f z r − d Ck Fr ,z ε 0 lnεmax/ε r & ε

Z Z∞ D k ﬁb d r = πω x,r, f ω d νbi (ω,x) Hν e r I N F 0 ×

R(x,0) Z Z Z r D k A z ,r f ﬁb z D − ψωe,x ω = D Hν R ψ y A ψ ,x y ψ ,x z ,r d µ y N R x,1 ω0 ,x ωe ωe I F ( ) norfFr e × − d r d Ck ψ ,x F ,ψ ,x z d νbi ω,x . ωe r ωe r ( ) CURVATURES OF SELF-CONFORMAL SETS 11

k The first limit formula formally depends on u via ψω,u inside π . Its advantage is less e ωe objects depend on the integration variable x. The second formula is better adapted to the dynamical system variable (ω,x) and has u replaced with x. We conjecture the uniform integrability assumption (3.0.1) cannot be weakened, see Remark 4.4. The proof is postponed after Remark 4.4. Future work will simplify the formula for special cases. For readers interested only in the Minkowski content, or in non-directional fractal curvature measures on d instead of d Sd 1: R R − × Corollary 3.4. (Non-directional version) Assume the situation of the theorem, except that f : d Sd 1 does not depend on its Sd 1 coordinate, and positive reach (Assumption 3.1) R − − is only× needed for k d 2. Then the limit (3.0.2) exists. That limit formula holds, and frac ≤ − fib R Ck (F, ) is proportional to µ, and f ω = f d µ. · In the self-similar setting, previous work used various definitions of A (Remark 4.12, [RZ10, Example 2.1.1]). We unify those approaches.

Corollary 3.5. Let every φi be a similarity, in addition to Assumptions 2.1, 3.1, 3.2. (Note

id, 1 therein.) Let A be as above, or the indicator function of any neighbor- ψ... = ψ0... = hood net from [RZ10, Example 2.1.1], or deﬁned any other way that makes Lemma 4.13 D true. Writing for the D-dimensional Hausdorff measure, G for the Haar probability ¬ H ¶ H on G : φ : i I , and c for the constants in (2.3.2), we have = i0 O d R ∈ ( )

c cR dist(x,X ) D Z Z Z r D k A z x ,r f ﬁb z d r frac − ( ) D Ck F, f = R | − | d Ck Fr ,z d (x), Hν A y z ,r d D y r F H F c cR dist x,φx 1X norfFr − H ( | ) Z Z ﬁb D 1 D f (z ,n) := (F )− f x, g n d (x)d G g . H H H F G

Remark 3.6. Arguably, the fractal curvature-direction measures should instead be nor- malized as

1 d Dk D k 1 frac d Dk − k Γ −2 + frac Cˆ F, : π 2− C F, ,(3.0.3) k ( ) = d− k d k k ( ) · Γ −2 + 1 · −¦ © Dk := min k ,dimMinkowskiF .

The motivation is, embedding a set K into a larger ambient dimension d does not cre- ate geometric information. Stronger yet, Zähle conjectured the fractal curvatures k > D never provide new information. This is proven in two cases: Fractal curvatures of an order k greater than the dimension of a “classical” set K simply repeat the highest avail- able order of curvature (due to the Steiner formula),

frac Cˆk (K , ) = Cmin k ,dim K (K , ) if reach K > 0. · { } · For a general, closed, Lebesgue null set K , the fractal Lebesgue measure repeats the next-lower measure, [RW10, RW12]:

frac d 1 frac d 1 (3.0.4) Cˆd K , S − = Cˆd 1 K , S − if Cd (K ) = 0. · × − · × CURVATURES OF SELF-CONFORMAL SETS 12

4. PROOFS 4.1. Exposing the dynamical system. The purpose of the next proposition is to rephrase the problem, of taking Cesaro limits of volume or curvature of parallel sets, into the lan- guage of the ergodic shift dynamical system (F,σ,ν). The geometric intuition is to study the curvature in a small ball around x F as the parallel set width r decreases. The conformal measure µ of this ball balances∈ out the rescaling factor. Instead of actual balls, we will use a tent function A supported by the ball (2.3.1). We introduce the extra in- R R tegrand 1 = Ad µ/ Ad µ into the curvature and use Fubini to make the “dynamical” measure µ accessible. The analytic intuition is to convolute the curvature measure with an integrable bump function A. If we had a sensible (harmonic analysis) group struc- ture on the fractal, it would preserve the metric. So we make the bump x A ( x z ,r ) depend only on pairwise distances x z . Lacking a global group, we7→ normalize| − | the bump mass locally at each midpoint|x.− | This leads us to study the dynamical system’sintegrands ρ, π for the rest of this paper. Recall from (2.3.4), Z r D k A x z ,r k , − ( ) ρ s,±t x,r, f := 1(s,t ] (r ) f z R | − | d Ck± Fr ,z . ( ] A y z ,r d µy norfFr − Remark 4.1. Borel measurability of ρ and π as a function of r can be proved as in [RZ10, Lemma 2.3.1]. The mapping r Ck (Fr , ) is weak*-continuous at almost every r (see [Zäh11, Corollary 2.3.5]). The non-osculating7→ · Assumption 3.1 (2) is used here. Since Ck ψF r , agrees with a (limit of) pullback of some Ck ψF const r , on sets we integrate over,· we will see it is automatically measurable. · Proposition 4.2. Let k 0,...,d , 0, and f : V Sd 1 0, be any nonnegative, ε > − [ ) measurable function. Suppose∈ { Assumption} 2.1 governs× our→ self-conformal∞ set F . Its parallel sets Fr shall be regular if k d 1 (but only k d 2 for f : V [0, )), Assumption 3.1. Then we have ≤ − ≤ − → ∞

(4.1.1) εmax 1 Z d r Z Z∞ 1 d r D k (k , ) f z r d C Fr ,z ρ x,r, f d µ x . − k± = (ε,ε±max] ( ) lnεmax/ε r lnεmax/ε r ε F 0

Proof. At each point z Fr we may introduce an arbitrary factor ∈ R A ( x z ,r )d µ(x) (4.1.2) 1 = R | − | . A y z ,r d µy R − Lemma 4.13 (1), (3) makes sure A d µ > 0 and A 0. Next, we may apply Fubini’s theorem because we are dealing with positive integrands≥ and ﬁnite measures. If ε < r εmax: ≤ R Z Z A x z ,r d µ x D k D k ( ) ( ) r − f z d Ck± Fr ,z = r − f z R | − | d Ck± Fr ,z A y z ,r d µy norfFr norfFr − Z Z r D k A x z ,r Z − ( ) (k , ) f z d C Fr ,z d µ x ρ x,r, f d µ x . = R | − | k± ( ) = ε,ε±max ( ) A y z ,r d µy ( ] J J norfFr − CURVATURES OF SELF-CONFORMAL SETS 13

Finally, we integrate over r 1d r and apply Fubini again. − 4.2. Dualizing the dynamics to apply the Birkhoff theorem and leave the distortion behind. Here we tell the story arc of this paper and treat the dynamics. The curvature Cesaro average is split into chunks mapped onto each other up to distortion of the un- derlying fractal. The local geometric covariance matches up only their curvature measures. To compensate, the dynamical system pulls back the test function. Asymptotics of the distortion exist only if the ordering of the conformal maps φi is reversed. The Perron-Frobenius operator achieves this, but requires the two-sided extension I N F of the code space. (Intuitively, summing over all possible n-letter words is the same× as summing over reversed words.) It also separates the distortion inside the measure from the dynamics map inside the test function. A vector (Kojima) Toeplitz theorem moves the curvature measures out of the Cesaro average. The Birkhoff theorem for the left shift ﬁts the Cesaro-averaged, pulled-back chunks of test function after compensating the following: The curvature lives on a parallel set “beside” the dynamical system F . This selects a different element from the covariance group element that transforms the normal directions. This work essentially reduces the self-conformal case to similarities, at the price of extra distortion. The distortion and the geometry packaged in ρ, π will later be treated 1 point-wise using the right shift σbi− . From a dynamics point of view, the different roles of left and right shift explain why the conformal measure µbi integrates the test function frac in the limiting Ck . Recall ρ from (2.3.4), π from (2.3.5) and (2.3.6).

Proposition 4.3. (core of proof) Assume k 0,...,d , f : V Sd 1 continuous and − R F is self-conformal (Assumption 2.1). Let∈ its { parallel} sets F×r be regular→ if k < d (only k < d 1 if f : V R) (positive reach Assumption 3.1). Let the expression − →  1 X § ª (4.2.1) k ,var x,r,1 1 r u k ,var x,r,1 εmax ρ ( ) + φ > ε π ( ) ln (max R(x,0),ε ,εmax] ωÞ0 n ωÞn ε { } n N | | ∈ be -uniformly r 1d r d ,x -integrable. Then for any continuous f : V Sd 1 , ε − µbi (ω ) − R the limit × → ε Zmax Z 1 D k d r (4.2.2) lim r − f z d Ck Fr ,z ε 0 lnεmax/ε r & ε d Sd 1 R − × Z Z∞ D k ﬁb d r = πω x,r, f ω d νbi (ω,x) Hν e r I N F 0 × exists.

Proof. Due to linearity, we may assume 0 f 1. First, we will rewrite (4.2.2) for any ﬁxed≤ .≤ To prevent that the r 1d r -integral eval- ε − + uates to , we will work with Ck and Ck− separately at ﬁrst. Proposition 4.2 below uses Fubini’s∞ − ∞theorem to makes µ appear in the next line,

εmax 1 Z Z d r 1 Z Z∞ d r L : f z r D k d C F ,z k , x,r, f d x . = εmax r = εmax ρ ± µ( ) ln − k± r ln (ε,εmax] r ε ε ε F 0 norfFr CURVATURES OF SELF-CONFORMAL SETS 14

The sequence sequence n R (x,n) is strictly falling (Lemma 4.11 (3) (4)). We can split the d r integral along it without7→ reversing the new bounds of any chunk of the integral. R R(x,n) The indicator function corresponding to the bounds is pulled into the subscript R(x,n+1) of ρ, see (2.3.4):

k , Z Z∞ ρ x,r, f 1 r d r (max± R(x,0),ε ,εmax] X > ε k , L { } ρ x,r, f d µ x . = ln εmax + ln{ εmax } (R±(x,n+1),R(x,n)] r ( ) ε n N ε F 0 ∈

1 k , k , n − Next, Lemma 4.14 below asserts ρ x,r, f π σ x,r φ u , f φx n . (R±(x,n+1),R(x,n)] = x n± x0 n | | ◦ | The geometric meaning is we preimage it under φx n and use the covariance and locality properties of curvature. Hence, |

  Z Z∞ X 1 r > ε k ,  r  d r L π σn x, , f φ d µ x . = + { εmax } x n±  x n  ( ) ln  |  r ··· n N ε | φ u ◦ F 0 x0 n ∈ | Temporarily, we pull the formally inﬁnite sum out of both integrals. Each summand integral exists due to positivity (or uniform integrability once proved),   Z Z∞ Z Z∞ X 1 r > ε k ,  r  d r L π σn x, , f φ d µ x . = + { εmax } x n±  x n  ( ) ln  |  r ··· n N ε | φ u ◦ F 0 F 0 x0 n ∈ |

By passing to the two-sided code space and to the invariant measure νbi = pµbi, we can apply the shift operator, σn ω,φ x σ n ω,x , (2.1.3). It consistently replaces x ωÞn = bi− ( ) with φ x. But left and right shifts| are inverse operations. Thus it replaces the original ωÞn | σn x with σn φ x x and x n with φ x n ω n: ωÞn = ωÞn = Þ | | | | |   Z Z∞ Z Z∞ X 1 r > ε k ,  r  d r L = + { } π ± x, , f φω n  d νbi (ω,x). εmax ωÞn Þ ··· p x ln  ◦ |  r n N φω n ε | φ u F 0 I N F 0 Þ ωÞ0 n ∈ | × |

We substitute r / φ u r in the d r integral and put the sum back inside, ωÞ0 n | 7→  § ª  ε Z Z∞ 1 φ0 u >  X ωÞn r k ,  d r L =  + | π ± x,r, f φω n  d νbi (ω,x). εmax ωÞn Þ ··· p x ln ◦ |  r n N φω n ε | I N F 0 Þ ∈ | × Because we want to move the ε-limit inside both integrals, we must assume the last line is uniformly integrable. The bounds K 1 p K and f φ 1 simplify that to our − ωÞn stated assumption (3.0.1). ≤ ≤ ◦ | ≤ Uniform integrability of L implies integrability. So we can subtract our chain of equa- + k ,+ k , tions for C and C , i.e., π π − . Then we draw the limit ε 0 into the integrals. k k− ωÞn ωÞn | − | → CURVATURES OF SELF-CONFORMAL SETS 15

We have proved, assuming the limit inside the double integral exists: (4.2.3) εmax Z Z Z Z∞  k , 1 d r ρ max± R x,0 , , x,r, f lim f z r D k d C F ,z lim ( ( ) ε εmax] εmax − k r =  { εmax} ε 0 ln r ε 0 ln p (x) → ε → ε ε N 0 norfFr I F ×  1 X § ε ª f φ n d r 1 u k x,r, ωÞ  d ,x . + εmax φ > π ◦ | νbi (ω ) ln ωÞ0 n r ωÞn   r ε n p φ x N | | ωÞn ∈ | The rest of this proof will show the integrand limit (4.2.3) exists. Clearly, k , can be ρ... ± ignored, k ε 1 ρ x,r, f max − (max R(x,0),ε ,εmax] lim ln { } = 0. ε 0 ε p (x) → § ª The Markov time N r /ε : max n : φ u > ε will help reinterpret the sum ( ) = N 0 r ∈ ωÞn as Cesaro average. The integrand of (4.2.3) becomes|    N r /ε 1 X f φ n lim0 ( ) k x,r, ωÞ . + r π ◦ | ε 0  lnεmax/ε N ωÞn   → ε n N r | p φ n x ( ε ) ωÞ ≤ | Lemma 4.6 below allows us to replace the following subexpression with its limit, N r /ε D ( ) . lnεmax/ε −→ε 0 Hν → n That eliminated the only ε outside N (r /ε). Recall φ u s , smax < 1. Thus ε 0 ωÞ0 n max | ≤ → implies N (r /ε) . We replace limε 0 with limN . Inside the integrals in (4.2.3), that leaves → ∞ → →∞   D 1 f φ X k  ωÞn  0 + lim π x,r, ◦ | . H N N ωÞn   ν n N p φ x →∞ | ωÞn ≤ | If αn is a weakly converging sequence of signed measures and bn a sequence of 1 P equicontinuous functions, then the Cesaro average N n N α (bn ) has the same lim- 1 P ≤ ∞ iting behavior as N n N αn (bn ). In other words, any converging measure αn may be replaced by its limit α ≤, see Lemma 5.1 below. Let us verify the assumptions: Lemma ∞ k 4.16 below provides the geometric justiﬁcation that αn ( ) := π (x,r, ) does converge. ωÞn The expression · | ·

f φ z ﬁb 1 X ωÞn f z = lim ◦ | ω N N n N p φ x →∞ ωÞn ≤ | will be lim 1 P b . It converges uniformly in z V Sd 1 for almost all ,x : N n N n − (ω ) Lemma 4.5 below≤ provides the reduction to the individual∈ × ergodic theorem (and a formula). The family of all φ-translates of z f z is equicontinuous. So we may replace πk with πk in (4.2.3). We have achieved7→ our goal to show the integrand limit exists: ωÞn ωe |   f φ ρ ...,εmax 1 N X ωÞn D lim ( ] k x,r,  k x,r, f ﬁb . εmax + π ◦ | = π ε 0 ln p N lnε /ε ωÞn   H ωe ω ε max n N p φ x ν → | ωÞn ≤ | CURVATURES OF SELF-CONFORMAL SETS 16

Remark 4.4. A converse shows the uniform integrability condition (4.2.1) likely cannot be improved. Without it, the same proof yields for continuous f 0, ≥ Z Z∞ Z Z∞ 1 k d r D k fib d r liminf ρ x,r, f d µ x π x,r, f d νbi ω,x . εmax (ε,εmax] ( ) ω ω ( ) ε 0 ln r Hν e r → ε ≥ F 0 I N F 0 × Both equality holds and the lower limit exists as a proper, finite limit, if and only if (4.2.1) after replacing ρvar by ρ and πvar by π is uniformly integrable. This follows from a converse to Fatou’s lemma ([Doo94, Chapters| | 6.8 and 6.18]). Assume we may replace Ck with Ck± in the continuity assertion of Fact 4.15 below when proving (4.6.1) (i.e. limn αn ). Then the last inequality improves to (4.2.4) εmax Z Z Z Z∞ 1 d r D k , d r liminf r D k f z d C F ,z x,r, f fib d ,x . εmax − k± r πω± ω νbi (ω ) ε 0 ln r Hν e r → ε ≥ ε N 0 norfFr I F × Both equality holds and the lower limit exists as a proper, finite limit, if and only if (4.2.1) after replacing var by and var by is uniformly integrable. (This was proved for ρ ρ± π π± self-similar F in [BZ11, Proposition 3.12].) Proof. (of Theorem 3.3) Lemmas 4.17, 4.18 derive uniform integrability with u (3.0.1) from the other conditions with u in Assumption 3.2. Then Proposition 4.3 above guar- antees convergence to the first limit formula (with u ). In a second run, we replace the constant u by the integration variable x in all the proofs (except in Lemma 4.9, where it plays a different role). Again, the other conditions without u imply uniform integrability without u and convergence to the second formula (without u ). It remains to show both formulas are equal. Write πk ,u for πk , and πk ,x when all instances of u in its definition (2.3.6) are re- ωe ωe ωe placed with x. This is the second limit formula’s integrand in the theorem. Define πk ,u , ωßm k ,x | π accordingly for finite words ωßm. Once as stated and once with u replaced, (4.5.4) ωßm implies| | k ,u k 1 k ,x π x,r, f = ρ ... φ x,r φ u , f φ− = π x,r / ψ x , f . ωßm ( ] ωßm ω0ßm ωßm ωßm ω0ßm,u | | | ◦ | | | Then as m , Lemma 4.16 gives πk ,u x,r, f πk ,x x,r / ψ x , f . This and the ωe = ωe ω0e,u → ∞ coordinate transformation ω,x,r ω,x,r / ψ x turn both limit formulas into ( ) ω0 ,u 7→ e each other.

fib The point η,u , the Brin group Gηu , and f ω were defined in (2.1.9). Recall Z Z orth 1 orth fib − f ω (z ,n) := f yˆ, ψη0 ,u yˆ gˆ ψω0 ,u z n d G gˆ d µ yˆ e e H I N F Gηu × does not depend on η,u . To readers not interested in directed curvature, the next lemma merely recalls Birkhoff’s theorem. CURVATURES OF SELF-CONFORMAL SETS 17

Although the dynamical system map φ is contractive in V , the limit f fib can de- ωßm ω pend on z V via the orbit of n Sd 1. | − ∈ ∈ Lemma 4.5. For almost all (ω,x), the expression f φ z ,n 1 X ωßm ( ) fib (4.2.5) lim ◦ | = f (z ,n), M M ω m M p φ x →∞ ωßm ≤ | converges uniformly in z ,n , i.e. w.r.t the norm on V Sd 1. It can be interpreted as the ( ) − × νbi-conditional expectation of the function a (ω,x,z ,n) on σbi-orbits at the point (ω,x,id), where orth 1 a ,y,z ,n ω,x, g : p x f x, g ψ z n . (τ ) = ( )− τ0e,y The limit simplifies to a constant in case f does not depend on the normal variable n Sd 1, ∈ − Z fib f ω (z ,n) = f d µ.

N Proof. Recall the extended dynamical system I F O (d ),σbi,νbi , νbi = νbi O(d ), × × ⊗H orth m m σbi ω,x, g σ ω,φ x, φ x g . − = ωßm ω0ßm | | We will rewrite f φ asymptotically in terms of this system. The base point z can ωßm be moved to x at the◦ price| of an additional ψ (deﬁned in (2.1.7)): orth f φ z ,n f φ z , φ z n ωßm ( ) = ωßm ω0ßm ◦ | | | orth orth f φ z , φ x ψ z n . = ωßm ω0ßm ω0ßm,x | | | The asymptotic approximation of f φ will be later obtained from p ωßm ◦ | orth orth f φ x, φ x g ψ z n ωßm ω0 m τ0e,y m ß (4.2.6) a σ ω,x, g | | (τ,y,z ,n) bi− = ◦ p φ x ωßm | by setting g : id, τ,y : ω,x . Indeed, the estimates φ x φ z s m diamV = = ( ) ωßm ωßm max | − | ≤ mγ and ψω0 ,x z ψ0 z smaxcψ depend only on m. Since f is uniformly continu- e − ωßm,x ≤ ous, this implies uniform| convergence   1 f φ (z ,n) X  ωßm m  (4.2.7) ◦ | a ω,x,z ,n σbi (ω,x,id) 0, M  ( ) −  M m M p φ x − ◦ −→ ωßm →∞ ≤ | i.e., at a rate that does not depend on ω,x,z ,n. Next, we will apply the Birkhoff ergodic theorem to a. Let be a countable, dense set of ,y,z ,n I N F V Sd 1. On each element of , theD limit τ − ∈ × × × D 1 X m (4.2.8) lim a τ,y,z ,n σbi ω,x, g M M ( ) − →∞ m M ◦ ≤ CURVATURES OF SELF-CONFORMAL SETS 18

exists for almost every ω,x, g . Lemma 4.9 provides a formula (a νbi-conditional expectation of a on σ -invariant sets). Once we know it converges uniformly in (τ,y,z ,n) bi τ,y,z ,n and g at each fixed (ω,x), we will be allowed to set τ,y = (ω,x) and g = id. Then (4.2.7) and (4.2.8) will yield the assertion. The auxiliary function orth τ,y,z ,n, g g ψτ0 ,y z n 7→ e is uniformly continuous, see Proposition 4.8 (1) for ψ z . We insert this into the ex- τ0e,y plicit formulas (4.2.6) for a m and (2.1.10) for f fib. So both (4.2.8) and f fib are uni- σbi− ω ω ◦ formly equicontinuous w.r.t. τ,y,z ,n and g at each fixed (ω,x). Equality extends to the closures τ,y,z ,n , g O (d ). The Arzela-Ascoli theorem improves point-wise convergence on O (d∈)Dto the∈ uniform convergence used above. We have shown D × f φ z ,n 1 X ωßm ( ) ◦ | M M m M p φ x −→ ωßm ≤ Z Z | orth 1 orth − a (ω,x,z ,n) ϑˆ,yˆ, ψη0 ,u yˆ hˆ ψη0 ,u x id d G hˆ d νbi ϑˆ,yˆ . e e H I N F Gηu × 1 1 Finally, p φ x − p φ u − asymptotically vanishes for any x,u V be- ωßm ωßm cause φ contracts| and−p is Hölder| continuous. So the left side the same for∈ almost ωßm all values| of x. Since the right side is continuous, it does not depend on the choice of x at all. Set x : u and simplify ψ u id, νbi ν lacking ϑˆ, ν/p µ. = 0ηe,u = = = R Recall (2.3.3): Hν : D ln φ σx d ν x . = F x0 1 ( ) − Lemma 4.6. Writing ε § ε ª

N ω,u , := max n N : φ0 u > , r ωÞn r ∈ | we have lnεmax/ε Hν lim = ε 0 N (ω,u ,ε/r ) D → for νbi-almost all ω and all u F. ∈ Proof. See the first half of the proof of [Pat04, Lemma 4.7] for convergence and identifi- cation of the limit, and e.g. [PU10, Thm. 1.9.7] for the entropy interpretation. Bounded distortion extends it to all u F . ∈ 4.3. Time-averaged distance, distortion, and ergodic fibres. The iterated function system maps F onto some smaller building block φω n F of the fractal under the non- | linear contraction φωn φω1 = φω n . But curvature and parallel sets transform ◦ ··· ◦ | nicely under a similarity φω0 n x. To compensate, the (non-contracting) distortion ψω n,x magnifies by one and contracts| by the other map. It usually does not converges| as n for an infinite sequence ω I N. This is why curvature densities in the sense of →[RZ10, ∞ BZ11] do not exist. Following∈ [Zäh01], the solution is to reverse the sub- words, ωÞn := ωn ωn 1 ...ω1. The two-sided right shift dynamical system inverts the − sub-words:| σ n ω,x σn ω,φ x . Hölder techniques will show ψ does con- bi− ( ) = ωÞn ωÞn,x | | verge. Our lim ψ y is equal to Zähle’stime-averaged distance function dist x,y . n ωÞn,x ω | CURVATURES OF SELF-CONFORMAL SETS 19

It is well-known the two-sided shift dynamical system I N F has an ergodic Gibbs measure νbi. But the directional nature of our curvature requires× a further extension by the orthogonal group. We provide an explicit formula for its ergodic ﬁbre decomposition. A key insight, the distortion derivative arises as a skewing factor, by which the ψ0 d ﬁbres differ from a product measure. (Readers interested in curvature on R without the normals can ignore and Lemma 4.9.) ψ0 Recall (2.1.7), 1 ψ y : φ x − φ y φ x , ωÞn,x = ωÞ0 n ωÞn ωÞn | | | − | Notice

φ y 1 τ0 (4.3.1) K − ψτ0 ,x y = K x ≤ φτ0 ≤ 1 and ψω n,x y = ψ0ω n,u x − ψω n,u y ψω n,u x . | | | − | n n Fact 4.7. A sequence a n is said to convergence at the rate cψs if supk a n a n+k cψs n n for all n. Note that if a n and bn converge at the rates ca s and cb s ,| then− the sequences| ≤ n n a n bn and 1/a n converge at the rates cab s and c1/a s respectively, with cab a 1 cb + 2 ≤ | | b1 ca + 2ca cb s and c1/a ca /infn a n . | | ≤ | | Proposition 4.8. (Distortion converges) For all x,y V¯ and all codes ω I N, (1) Both ∈ ∈

ψω,x y : lim ψ y, e = n ωÞn,x →∞ | ψ y lim ψ y ω0e,x = n ω0Þn,x | nγ →∞ converge at the universal rate cψsmax. The constant 0 < cψ < does not depend on x,y,ω. Thus both limiting functions are uniformly continuous∞ in ω,x,y . (The derivative of the limit ψω,x agrees with the limit of the derivative ψ .) e ω0Þn,x | (2) There is a constant Kψ such that

φ y φ x ωÞn ωÞn const | − | const.

≤ φ x y ≤ ωÞ0 n | − 1 γ (Here, we made use of the C + conformal extension of φi to VMU , subject to (2.1.6).)

 1 Then φ x − φ y φ x = φ y φ x / φ x and bounded distortion ωÞ0 n ωÞn ωÞn ωÞn ωÞn ωÞ0 n | − | | − | (2.1.6) complete| its proof. |

Next, we will show the ﬁrst assertion. Convergence of ψτ,x y is shown in [Zäh01, Theorem 1.(i)]. Though stated only for x,y X, the proof works on all of V . It could ∈ CURVATURES OF SELF-CONFORMAL SETS 20

be modiﬁed to account for pairwise distances, but for ψ y we need an approach 0τ,x inspired by cross ratios. The equality

1 ψτ,u v = φτ0 u − φτu φτv − permits us to rewrite

φ y φτy φτz / φ z φ z ψτ,z y ψτ,x z τ0 τ0 τ0 (4.3.2) ψτ0 ,x y = = − = x z y y x z x φτ0 φτ φτ / φτ0 φτ0 ψτ,y ψτ,z − in terms of quantities which are known to converge individually at the desired rate. It remains to find uniform bounds on all the factors. The as-yet-undetermined variable z was introduced to bound the denominators away from zero: since V is connected, we can always find some z V such that both x z and y z exceed diamV /4. Our ∈ | − | − second assertion then places bounds on the ψτ,v u -like expressions. At the price of enlarging the constant cψ, this gives us both the asserted (rate of) convergence of, and bounds on ψ y . 0τ,x orth The next goal is convergence of y . Each of the finitely many functions ψ0τ,x orth 1 x φi0 x is γ-Hölder continuous. Since infi ,x φi0 x > 0, φi0 x = φi0 x − φi0 x also is7→ Hölder with respect to the action invariant metric on O (d ). Application of the triangle inequality, action invariance, and Hölder continuity let us demonstrate the Cauchy sequence property (suppressing “orth”):

1 1 − − dO(d ) φ0 x φ0 y , φ0 x φ0 y ωän+k ωän+k ωÞn ωÞn | | | | n k 1 + 1 1 X− − − dO(d ) φ0 x φ0 y , φ0 x φ0 y ωãi +1 ωãi +1 ωgi ωgi ≤ i =n | | | | n k 1 + 1 X− dO d φ φ x − φ φ y ,id ( ) ω0 i +1 ωgi ω0 i +1 ωgi ≤ i =n | | n+k 1 X− i γ nγ constsmax const0 smax. ≤ i =n ≤

By multiplying this with ψ y , we have proved ψ y itself converges at the desired 0τ,x 0τ,x rate. Since ψτ,x x = 0 and V is bounded and connected, we can represent ψτ,x y as an integral of along a curve from x to y to see it converges as asserted. ψ0τ,x

Recall the shift dynamical system on I N F O (d ) is given by νbi = νbi Haar and × × ⊗ orth 1 n n n − σbi ω,x, g = xgn.ω, σ x, φx0 n σ x g . | | Let η,u I N F be the reference point, and recall ∈ × orth 1 orth orth Gηu ψ x − φ x ψ φi x : x F,i I . = η0e,u i0 η0e,u ∈ ∈ O(d ) CURVATURES OF SELF-CONFORMAL SETS 21

Lemma 4.9. (Ergodic ﬁbers of group extended shift) Let f : I N F O (d ) R be measurable and νbi-integrable. The Birkhoff limit is almost everywhere× given× by→

1 X n (4.3.3) lim f σbi ω,x, g = N N →∞ n

Proof. Firstly, we will summarize the theory of Brin groups according to [Dol02, Section 2.2] because that work does not assume a smooth manifold. The shift space I Z,σbi,νbi can be extended by a Hölder function τ : I Z H with values in a compact, connected → Lie group H. The skew product T w, g = σbiw,τ(w ) g defines the extended dy- Z n 1 namical system is I G ,T,νbi HaarH . Write τn (ω,x) := τ σbi− w ...τ(w ) and 1 ,x for its point-wise× inverse⊗ in H. The stable sets, unstable sets, and orbits are τ−n (ω ) defined by § ª s 1 W w, g = w , lim τ−n (w )τn (w ) g : wi = w i ,i N for some N , n ≥ § →∞ ª u n 1 n W w, g w , lim τn σ w τ σ w g : wi w i ,i N for some N , = n bi− ( ) −n bi− ( ) = →∞ ≤ − o ¦ n n 1 n © W w, g = σbi (w ),[τn (w )] g , σbi− (w ), τ−n σbi− (w ) g : n N . ∈ Z A t-chain is a finite sequence W = w(1) g (1),...,w(n) g (n) from I H with w(i +1), g (i +1) s u o × ∈ W w(i ), g (i ) W w(i ), g (i ) , and an e-chain also allows W w(i ), g (i ) . The holonomy 1 element of W ∪is g W g n g ; it depends only on the I Z part in case of a t-chain. The ( ) = ( ) (−1) Brin ergodicity group Γe (w ) H at w is generated set-theoretically by all e-chains that start and end in w . ⊆ The system is called reduced if any two points in I Z can be connected by a t-chain W with trivial holonomy element g (W ) = id. Then Γe is generated set-theoretically by all e-chains without any restrictions. Its closure Γe H is generated by all τ(w ), w I Z. ⊆ ∈ The extended dynamical system is ergodic if and only if H = Γe . Our goal is to reduce it. An active coordinate transformation Φ of I Z H, × Φ w, g = w,α(w ) g = w 0, g 0 conjugates T to T (i.e., T T ) if and only if it maps to 0 Φ = 0 Φ τn ◦ ◦ n 1 τ0n w 0 = α σbiw τn (w )α− (w ). Thus g W w g W 1 w . Denote u ,v the local product, i.e., u at neg- 0 ( ) = α( end) ( )α− (1) [ ] ative and v at positive indices. We pick a reference point w 0, connect it to w by the t-chain W w,w 0,w ,w 0, and set w : g 1 W , i.e., w = α( ) = − ( w ) 1 0 n 0 1 n 0 α w lim τ w τn w ,w lim τn σ w ,w τ σ w . ( ) = n −n ( ) n bi− −n bi− →∞ →∞ Our coordinate transformation Φ reduces the system because g (Ww ) = id. Therefore, 0 the dynamical subsystem I Z e ,T ,νbi is ergodic, where is the Haar prob- Γ 0 è Γe × ⊗ H H ability on Γe . Secondly, we will more explicitly compute the ergodic fibres of the original dynamical system. (Our α corresponds to h of [PP97, Theorem 3] and u of [Rau07, Theorem 1.3] CURVATURES OF SELF-CONFORMAL SETS 22 and [CR09, (10)], in the sense that Γe α agrees with its counterparts.) In the original coordinates, Γe and α depend on the choice of reference point: w w 1 w : w I Z . Γe = α(σbi ˆ) τ( ˆ) α− ( ˆ) ˆ H ∈ For all h H, define the function ∈ f h w, g := f w, g h on I Z H. Birkhoff’s ergodic theorem for the ergodic, reduced subsystem I Z Γe implies 1 that f×h converges on -almost all α w g e : × Φ− Γe ( ) Γ ◦ H ∈ 1 X l 1 X 1 l f T w, g h = f h Φ T Φ w, g n n − 0 n l

Lemma 4.11. (Renewal radii) For µ-almost all x,y F , all m 0, z V: ∈m ≥ ∈ (1) Avoids overlap sets φτF r φϑ F r (τ,ϑ I ): ∩ ∈ z x < aR x,m z / φ X c , ( ) = x m R(x,m) | − | ⇒ ∈ | y x < 2aR (x,m) = x m = y m, − ⇒ | |

(2) Covariance: R x,m φ σm x R σm x,0 , ( ) = x0 m ( ) (3) Monotonicity: a 1 dist F,V| c R x,m R x,m 1 0, − ( ) εmax ( ) ( + ) > (4) Limit: R x,m 0, ≥ ≥ ≥ ( ) m R −→ →∞ (5) Integrability: lnR y,0 d µy < and, for some c > 1: F ∞ Z (4.4.1) c lnR(y,0) d µy < . | | ∞ F

c Proof. (Compare [Pat04].) The ﬁrst assertion holds if any y V avoids φx m X if y x < ∈ | − 2aR (x,m). Indeed: given z , set y := 2z x. Given y F , (OSC) implies y φx m F . We use the Open Set Condition, the− triangle inequality,∈ bounded distortion∈ | (2.1.6) and an undetermined 0 < cR 1: ≤ c (OSC) c c dist y,φx m X = sup dist y,φx n X sup dist x,φx n X x y | n m | ≥ n m | − − ≥ n ≥ c > sup dist φx n σ x,φx n X 2aR (x,m) n m | | − ≥ (2.1.6) 2a K 1 sup φ σn x dist σn x,X c R x,m : 0. φ− x0 n ( ) ( ) = ≥ n m | − cR ≥ We solve the last equation for R (x,m),

cR R x,m sup n x dist n x,X c , ( ) = φx0 n σ (σ ) 2a Kφ n m | ≥ and recuperate the original deﬁnition of R (x,m) if cR 1 is the largest value such that ≤2 maxx F R (x,0) εmax. (As an aside, Re (x,0) R (x,0) KφRe (x,0).) The∈ second≤ assertion (covariance) follows≤ from, (n≤ m) ≥ n m n m m φ σ x φ σ x φ m σ σ x , x0 n = x0 m (0σ x) n m − ( ) | | | − CURVATURES OF SELF-CONFORMAL SETS 24 and the deﬁnitions of R (x,m) and R (σm x,0). Monotonicity is trivial except for R (x,m) > 0, which follows from the integrability. The limit assertion stems from

R x,m m x R m x,0 s m . ( ) = φx0 m σ (σ ) maxεmax | ≤ lnR y,0 c Integrability of c ( ) is reduced by c dist y,X = Re y,0 R y,0 εmax to | | 0 Z ≤ ≤ c (4.4.2) c ln dist(y,X ) d µy < . | | ∞ R [Pat04, Lemma 3.8] proves ln dist( ,X c ) d µ is ﬁnite using (SOSC). Our case is similar. Patzschke dominates the integral| by· a geometric| series. We can safely increase its base

 while (in Patzschke’s notation) 0 < lnc < ln C 1 S r ln 1 C 1ψη . 2− 0η min 5− −

The purpose of A is to sample the curvature located in a small ball B (x,ar ), r < R (x,m). It has to work together with weak convergence of curvature measures (Section 4.6). So we deﬁne it as a continuous counterpart of 1B(x,ar ) (z ): the tent function x z A ( x z ,r ) := max 1 | − | , 0 . | − | − ra Many other choices are viable due to (4.1.2). So we will treat A as a black box and de- mand only the next lemma holds.

Remark 4.12. Our A uniﬁes previous approaches in the self-similar setting. It can be re- [RZ10] placed with any of the locally homogeneous neighborhood nets A F (x,r ) from [RZ10,

Example 2.1.1 if A x z ,r : 1 [RZ10] z . (Our continuity is replaced with their ] ( ) = A F (x,r ) ( ) Lemma 2.1.3.) This includes| − | a net that eliminates the denominator A in ρ and π.

Items 1, 3 are designed to enable Fubini (Proposition 4.2); 2, 4, 5 to localize and preimage curvature (Lemma 4.14); and 3, 6 to preserve weak convergence (Lemma 4.16).

Lemma 4.13. (Covariant neighborhood net) For almost all x,xˆ,y,yˆ F , all u ,zˆ,z V, ∈ ∈ and almost all r,rˆ εmax: ≤ (1) (Measurability) (x,z ,r ) A ( x z ,r ) is Borel measurable; 0 A ( x z ,r ) 1. (2) (Locality) 7→ | − | ≤ | − | ≤

¦ d © zˆ R : A ( xˆ zˆ ,rˆ) = 0 B◦ (xˆ,arˆ) V ∈ | − | 6 ⊆ ⊆ . (3) (D-set property) There is a constant CA , 0 < CA < , such that R ∞ A y z ,r d µy 1 ˆ ˆ ˆ ˆ CA− − D CA . ≤ rˆ ≤ More generally, let write I and z F . Then τ ∗ ψτ,u ψτ,u r ∈ ∈ R A ψ y ψ z ,r d µy 1 τ,u τ,u (4.4.3) CA− − D CA . ≤ r ≤ (4) (2-chain of supports avoids intersections φ F φ F ) If r R x,m , τ rˆ xˆ m rˆ ˆ (ˆ ) ∩ | ≤ A ( xˆ zˆ ,rˆ) = 0 and A yˆ zˆ ,rˆ = 0 = xˆ m = yˆ m. | − | 6 − 6 ⇒ | | CURVATURES OF SELF-CONFORMAL SETS 25

(5) (Covariance w.r.t the ψ-distorted distance; compatibility with the dynamics) If r R x,m and φ z φ F , then ˆ (ˆ ) xˆ m xˆ m rˆ | | ≤  ∈ 

 m rˆ  A yˆ φxˆ m z ,rˆ = A  ψxˆ m,u σ y ψxˆ m,u z , 1 yˆ m = xˆ m . − |  | − | φ u  | | x0ˆ m | (6) (Lipschitz continuity) As a function x z A ( x z ,r ) for ﬁxed r , | − 1| 7→ | − | Lip(A) r − . ≤ Proof. Point one and six are obvious. The second and fourth follow from Lemma 4.11 point (1).

The third assertion will be proved in the version with ψτ,u . It contains the simpler version for τ = ∅, ψτ,u = id. Abbreviate ψ := ψτ,u . We want to estimate 1 Z Z Z ¦ © A ψy ψz ,r d µ y = 1 1 t A ψy ψz ,r d µ y d t − − ≥ − F 0 F 1 Z Z ¦ © = 1 t ar ψy ψz d µ y d t . ≥ − 0 F

There is a point xz F such that ψz ψxz r . The triangle inequality implies ∈ − ≤ ψy ψz r ψy ψxz ψy ψz + r, ¦ − − © ≤ ¦ − © ≤ ¦ − © t ar + r ψy ψxz 1 t ar ψy ψz 1 t ar r ψy ψxz , ≥ − ≥ ≥ − ≥ − ≥ − and then Proposition 4.8 (2) ¦ © ¦ © ¦ © 1 (t a + 1)r Kψ y xz 1 t ar ψy ψz 1 (t a 1)r /Kψ y xz . ≥ − ≥ ≥ − ≥ − ≥ − We integrate w.r.t. d µy d t and then use the D-set property (2.1.4) of µ for balls cen- tered on xz , 1 1 Z Z Z D 1 D C F (t a + 1)r Kψ d t A ψy ψz ,r d µ y C F− (t a 1)r /Kψ d t . ≥ − ≥ − 0 F 1/a

D 1 D This inequality ﬁts between CA r and CA− r for a suitable CA . 1 To prove the ﬁfth assertion, we will use φ u − φx m ψx m,u up to an additive x0ˆ m ˆ = ˆ constant (Proposition 4.8 (3)): | ◦ | | (4.4.4) 1 h i 1 − m m φ u y φx m z φ u − φx m σ y φx m z ψx m,u σ y ψx m,u z . x0ˆ m ˆ = x0ˆ m ˆ ˆ ˆ = ˆ ˆ ˆ | − | | | − | | ◦ − | Inserting this into (2.3.1) results in the assertion. S 4.5. Localizing and scaling the curvature under the inverse IFS. As Fr = τ I m φτF r , parts of F agree with F . Under the similarity u , this is the image of another∈ par- r φτ r φτ0 allel set of ψτ,u F . For given m N and in a small area around x F , the agreeing r are less than R (x,m). This idea also∈ works for the associated curvature∈ measure instead of the set. Note ρ, π below contain the chunk of curvature between r R (x,m) and ≤ r R (x,m + 1). CURVATURES OF SELF-CONFORMAL SETS 26

(The curvature of F is a combination of those of 1 F , compare Ber03b . ψ r ψ− ψ r [ ] But this does not help us.) Note Z k , (4.5.1) ρ x,r, f qm id,x,z ,r f z d C Fr ,z , (R±(x,m+1),R(x,m)] = ( ) k±

Z k , (4.5.2) πτ ± x,r, f = q0 ψτ,u ,x,z ,r f z d Ck± ψτ,u F r ,ψτ,u z , and with a placeholder ψ, ! r r D k A ψx ψz ,r (4.5.3) q ψ,x,z ,r : 1 − . m = (R(x,m+1),R(x,m)] R D − ψ x y A y z ,r d y 0 ψ0 ψ ψ µ − Lemma 4.14. Given xˆ F , u V , m N, the substitutions ∈ ∈ ∈ m x := σ xˆ, τ := xˆ m, r := rˆ/ φτ0 u | transform π and ρ into each other:

(4.5.4) πk , x,r, f φ ρk , x,r , f . τ ± τ = (R±(xˆ,m 1),R(xˆ,m)] ˆ ˆ ◦ + Proof. Abbreviate ρ : ρk , x,r , f . For clarity, note that = (R±(xˆ,m+1),R(xˆ,m)] ˆ ˆ

x φ x, m τ , r r φ u . ˆ = τ = ˆ = x0ˆ m | | | In the ﬁrst half of the proof, we will rewrite ρ’s curvature density qm (id,xˆ,zˆ,rˆ) in terms of the new variables

y : 1y , r r u , z 1z . = φτ− ˆ = ˆ/ φτ0 = φτ− ˆ They will later leave the measure r 1d r unchanged. This means x, y , z , and r are preim- ˆ− ˆ ˆ ˆ ˆ ˆ aged m times under the IFS. R (xˆ,m) was deﬁned to be covariant:

¦ m m m m © 1 R (xˆ,m + 1) < rˆ R (xˆ,m) = 1 φτ0 σ xˆ R (σ xˆ,1) < r φτ0 u φτ0 σ xˆ R (σ xˆ,0) { ≤ } ≤   ( ) u φτ0  r  (4.5.5) = 1 R (x,1) < r R (x,0) = 1 R (x,1) < R (x,0) φτx ≤  ψ x ≤  0 0τ,u

In the denominator of qm (id,xˆ,zˆ,rˆ), Lemma 4.13 (4) tells us

yˆ m = xˆ m. | | Therefore, we can transform both instances of A with the same τ, see (4.4.4) in the same lemma: (4.5.6) A ( xˆ zˆ ,rˆ) = A xˆ φτz ,rˆ = A ψτ,u x ψτ,u z ,r 1 τ = xˆ m , | − | − − { | } A yˆ zˆ ,rˆ = A yˆ φτz ,rˆ = A ψτ,u y ψτ,u z ,r 1 τ = yˆ m . − − − | CURVATURES OF SELF-CONFORMAL SETS 27

(A geometric interpretation: the covariance of A was designed to match that of the curvature.) We integrate the last two lines. Then the Perron-Frobenius operator of µ intro- D duces d m d y . That is absorbed into , µ σ− / µ = φτ0 ψτ,u Z ◦ Z A y z ,r d y A m y z ,r 1 d y ˆ ˆ ˆ µ ˆ = ψτ,u σ ˆ ψτ,u τ=yˆ m µ ˆ − − { | } Z D = φτ0 y A ψτ,u y ψτ,u z ,r d µ y − Z D D (4.5.7) ψ y A ψ y ψ z ,r d µy φ u . = τ0 ,u τ,u τ,u τ0 − The remaining factor of qm (id,xˆ,zˆ,rˆ) transforms as D k D k D k rˆ − = r − φτ0 u φτ0 u − . We assemble both factors above and (4.5.5), (4.5.6), (4.5.7) to obtain k (4.5.8) φτ0 u qm (id,xˆ,zˆ,rˆ) = q0 ψτ,u ,x,z ,r . In the second half of the proof, we will make use of the locality (2.2.3) and scaling (2.2.2) properties of curvature measures to perform the remaining substitution zˆ 7→ ψτ,u z . First, we rename zˆ to φτz , Z ρ = qm (id,xˆ,zˆ,rˆ) f φτz d Ck± Frˆ,φτz ,

To localize, only a small area around xˆ supports zˆ qm (id,xˆ,zˆ,rˆ). This is because A 7→ lives on B◦ x,aR x,m (Lemma 4.13 (2)), which is disjoint from φ F whenever (ˆ (ˆ )) v m rˆ τ : x m v m I m (Lemma 4.11 (1)). Therefore, we may replace| F with φ F = ˆ = rˆ τ rˆ inside the| 6 curvature| ∈ measure: Z ρ q id,x,z ,r f φ z d C φ F ,φ z . = m ( ˆ ˆ ˆ) τ k± τ rˆ τ

1 Since 1 is a similarity with Lipschitz constant u , it maps parallel sets to ψτ,u φτ− φτ0 − parallel sets,◦ ψ φ 1 φ F ψ F ψ F . τ,u τ− τ rˆ = τ,u rˆ/ φ u = τ,u r ◦ τ0 We may introduce the similarity via the scaling property| | (2.2.2) of curvature measures, Z k ρ φ u q id,x,z ,r f φ z d C ψ φ 1 φ F ,ψ φ 1 φ z = τ0 m ( ˆ ˆ ˆ) τ k± τ,u τ− τ rˆ τ,u τ− τ ◦ ◦ Z k = φτ0 u qm (id,xˆ,zˆ,rˆ) f φτ z d Ck± ψτ,u F r ,ψτ,u z . ◦ (Scaling also proves Ck± ψτ,u F r , is measurable as a function of r on the set where we need it, see Remark 4.1) This combines· with (4.5.8) to complete the proof.

4.6. Curvature of distorted images converges. In the last section, we compared Fr to a (locally) ﬁxed range of parallel sets of the distorted fractal ψ F . Unlike the self- ωÞn,u similar case, we cannot eliminate the distortion ψ altogether.| But this map con- ωÞn,u verges as n and depends asymptotically only on| the “dynamical” variable (ω,u ) k , N → ∞ ( ) ∈ I F . In this section, we prove the associated “chunk of curvature” πx n± converges, too.× | CURVATURES OF SELF-CONFORMAL SETS 28

The next continuity result will be applied to K n : ψ F and K : ψ F for = ωÞn,u ∞ = ωe,u N | u F , ω I . If we had a corresponding result about the variations Ck±, our main theorem∈ would∈ hold for those, too (Remark 4.4). It seems difﬁcult to quantify the rate of convergence unless the reach can be controlled, compare [CSM06, Corollary 2, p. 384] in the smooth setting.

Fact 4.15. Let K ,K n d be a sequence of nonempty, compact sets, and suppose K n ∞ R converges to K in the Hausdorff⊆ metric, i.e., ∞ ¦ © lim inf r > 0 : K n K and K n K 0. n r ∞ r∞ = →∞ ⊇ ⊆ (1) Let k d 1,d . For all r > 0 except possibly a countable set, ∈ { − } wlim C K n , Sd 1 C K , Sd 1 . n k Ýr − = k gr∞ − →∞ · × · × (2) Let k 0,...,d 1 , and assume K has positive reach for a fixed r 0. Then gr∞ > n ∈ { − } KÝr has uniformly positive reach for any sufficiently large n, and wlim C K n , C K , . n k Ýr = k gr∞ →∞ · · Proof. The first assertion is due to Stacho ([Sta76, Theorem 3]). It was put in the present form by [RSM09, Theorem 5.1], [RW10, Corollary 2.5]. The second assertion can be found in [RSM09, Theorem 5.2] for (non-directional) Federer curvatures only. The same n n work shows liminfn reach KÝr > 0. This is enough for the normal current of KÝr to →∞ converge in the flat semi-norms ([RZ01, Theorem 3.1]). Since Lipschitz-Killing curvature has a current representation ([Zäh86, (18)]), it converges weakly. Recall q (4.5.3) is a template for the integrands of k , x,r, f (4.5.2) for finite I πτ ± τ ∗ and πk x,r, f (2.3.6) for infinite ω I N, ∈ ωe ∈ ! r r D k A ψx ψz ,r q ψ,x,z ,r 1 − . m = (R(x,m+1),R(x,m)] R D − ψ x y A y z ,r d y 0 ψ0 ψ ψ µ − Lemma 4.16. Let ,x I N F , z V Sd 1, fix a reference point u V , and assume (ω ) − r > 0 is regular in the sense∈ of× Assumption∈ × 3.1. ∈ (1) As a weak limit of signed measures,

k k (4.6.1) π x,r, wlimn π x,r, . ω ( ) = ω n ( ) e →∞ Þ · | · (2) The limit q0 ψω,u ,x,z ,r lim q0 ψ ,x,z ,r e = n ωÞn,u →∞ | converges uniformly in z (possibly except at two values of r ). There is even a constant cq > 0 such that it converges at the universal rate

1 1 c 1 dist r ψ x − , R x,1 ,R x,0 − r k 1s nγ . q + ω0 ,u ( ) ( ) − − max e { } Proof. First, we will prove q converges at the desired rate. It is enough to know every factor and its denominator are bounded and converge at such a rate (Fact 4.7). Both nγ ψ x and ψ x converge at the rate cψsmax (Proposition 4.8 item (1)). The factor ω0Þn,u ωÞn,u | | 1 − k 1 R x,1 ,R x,0 r ψ x r ( ( ) ( )] ω0Þn,u − | CURVATURES OF SELF-CONFORMAL SETS 29 is bounded by r k . Only at the endpoints of the interval can it be discontinuous. To see − how far its argument can stray,

1 1 1 1 cψ nγ r ψ x − r ψ x − = r ψ x − ψ x − ψ x ψ x r s . ω0Þn,u ω0e,u ω0e,u ω0Þn,u ω0e,u ω0Þn,u K 2 max | − | − | ≤ The bound r k originating from the supremum norm at the worst n determines its over- − 1 c r n ψ γ − all rate 2 k smax, where δ is the distance from r ψ x to the nearest endpoint K δr ω0Þn,u 1 | (which can be estimated by that of r ψ x − ). ω0e,u The next factor A ψ x ψ z ,r is trivially bounded by 1, and converges at ωÞn,u ωÞn,u | − | a rate r 1c s nγ due to Lip A r 1 (Lemma 4.13 (6)). The estimate (Lemma 4.13 (3)) − ψ max ( ) − ≤ R D ψ A ψ x ψ z ,r d µy 0 ωÞn,u ωÞn,u 1 1 | − | cψ− CA− D cψCA ≤ r ≤ makes sure we can divide without losing rate of convergence. Straightforward reasoning treats the last formula’s numerator. Convergence of q ψ ,x,z ,r is proved. 0 ωÞn,u | In preparation for the ﬁrst item, let µn µ be any sequence of weakly converging → ∞ measures, and g n g norm converging, continuous functions. Note supm µm < due to Banach-Steinhaus.→ ∞ Then ∞

(4.6.2) µn g n µ g µn µ g + sup µm g n g m − ∞ ∞ ≤ − ∞ ∞ − ∞ ∞

k permits us to treat the functions and measures separately. Back to µn := π x,r, f , ωÞn | we have proved its integrand g z : q ψ ,x,z ,r f z converges uniformly. n = 0 ωÞn,u Positive reach of F (Assumption 3.1 (3))| and Fact 4.15 (and equicontinuously ψωe,u r uniform convergence of ψ inside d z ) conﬁrm the curvature measure µ d z : ωÞn,u n = | Ck ψ F ,d ψ z converges. (The point-wise limit πω x,r, f remains mea- ωÞn,u r ωÞn,u e | | surable as a function of ω,x,r , see Remark 4.1).

4.7. Sufﬁcient conditions for the uniform integrability assumption. Our main result depends on a uniform integrability assumption. We can always prove it for Cd (Minkowski content) and Cd 1 (surface content) since these contain no principal curvatures. More generally, Lemma− 4.17 below carefully estimates a dominating function. Its resulting order of integrability is necessary in the self-similar case. The converse dominated ergodic theorem [Pet89, Theorem 3.1.16] shows this when applied to [BZ11, (3.4.1)] and the suspension ﬂow. Uniform integrability singles out µbi and νbi as the geometrically “correct” limit measures. Recall their thermodynamic potential is the contraction ratio x D logφx1 σx. Measures generated by arbitrary potentials are studied in multifractal formalism7→ − [Pat97]. At the price of an extra multiplier, these could replace µbi and νbi up to here. But different potentials make mutually singular measures. So fractal curvature can have a density with respect to at most one of them. CURVATURES OF SELF-CONFORMAL SETS 30

Lemma 4.17. The uniform integrability assumption (3.0.1) of the theorem holds if the function (4.7.1)   M 1 1 X−  r  h r,ω,x r k sup C var ψ F ,ψ B◦ x,ar 1 ( ) = − k ωßm,u ωßm,u ( ) (R(x,1),R(x,0)]   M >0 M r   m=0 | | ψ x ω0ßm,u | is in the Zygmund space, i.e.,

Z Z∞ d r max 0, h (r,ω,x) ln h (r,ω,x) d νbi (ω,x) < { | | | |} r ∞ I N F 0 × and if also Z Z 1 r C var F , B◦ x,ar ∞ (max R(x,0),ε ,εmax] ( ) k r ( ) d r sup { } d ν ω,x < . εmax k bi ( ) εmax ln r r 0<ε 2 ε ∞ I N F 0 ≤ ×

Note ψ x = 1 simpliﬁes h in case u is replaced with x, see Theorem 3.3. ω0ßm,x | Proof. We prepare to integrate a dominating function, i.e., the supremum of (3.0.1) over all ε (0,εmax/2]. First, we eliminate ε in favor of the summation range M of m. Besides M , only∈ this subexpression of (3.0.1) depends on ε explicitly: § ª § ε ª ε X X m M := 1 r φ u > ε 1 s > 1 + ln /lnsmax, ω0ßm max r r m N | ≤ m N ≤ ∈ ∈ 1 M 1 1 1 lnr lnε 1 c1 c2 lnr . M ln εmax M ln2 + lns ln − ln M ( + ) ε ≤ max εmax ε ≤ | | | | − r D k A ψx ψz ,r k ,var r − ( ) Secondly, we can estimate π ’s integrand 1 R x,1 ,R x,0 R D − . ω m ( ( ) ( )] ψ x ψ y A ψ|y ψz ,r| d µ y ß 0 0 ( ) ( ) Lemma 4.13 item (3) “cancels”| r D with the denominator.| Item| | (2)| replaces| − A| in the numerator with the interior of its supporting ball B (x,ar ):  

k ,var r   k var ◦ π (x,r,1) 1(R(x,1),R(x,0)]   CA r − Ck ψω m,u F ,ψω m,u B (x,ar ) . ωßm   ß r ß | ≤ ψ x | | ω0ßm,u |

The estimate K 1 ψ x K in the indicator function will help us ﬁnd a ﬁnite − 0 ≤ ωßm,u ≤ measure τ. We apply the| above preparations to (3.0.1) and obtain the dominating function:  § ª  k ,var ρ x,r,1 1 r φ0 u > ε  (max R(x,0),ε ,εmax] ( ) X M ωßm k ,var  sup  { } | π x,r,1  εmax + εmax m ( ) 0<ε ε /2  ln ln M ωß  max ε m N ε | ≤ ∈

 k ,var    ρ max R x,0 , , (x,r,1) 1 X sup ( ( ) ε εmax] sup c c lnr k ,var x,r,1  { ε}max  +  ( 1 + 2 )π ( ) ε ln M M ωßm ≤ ε m

  var ◦ 1(max R(x,0),ε ,εmax] (r )Ck Fr , B (x,ar )  { }  sup ε  + [CA (c1 + c2 lnr ) h (r,ω,x)]. ln max r k ≤  ε ε  | |

Thirdly, the Young inequality (known from the duality of Orlicz spaces) will be used to show the h-summand in the last line is integrable w.r.t r 1d rd ,x . (Recall − νbi (ω ) R (x,1)/K < r R (x,0) K .) The function CA (c1 + c2 lnr ) alone is “very” integrable: By (4.4.1), there is≤ a c3 > 1 such that | |

R(x,0)K Z Z c ln K Z lnr d r 3− lnR(x,0) c3| | d ν (x) c3− d ν (x) < . r ≤ lnc3 ∞ F R(x,1)/K F This also shows the measure d r, ,x : r 11 r d r d ,x is finite. τ( ω ) = − (R(x,1)/K ,R(x,0)K ] ( ) νbi (ω ) ¦ 2 t © We have concatenated r lnr with t Ψ(t ) := min t ,c3 , and Ψ( lnr ) stays integrable. The function Ψ is7→ convex. | | A straightforward7→ computation shows| its| Legendre R t transform t : inf u : u > s d s satisfies lim t lnt / t lnc . So h Φ( ) = 0 Ψ0 ( ) t ( ) Φ( ) = 3 Φ( ) →∞ is τ-integrable due to our{ assumption} on h. The Young inequality for Ψ and Φ (see| | [RR91]) yields Z Z Z lnr h (r,ω,x)d τ(r,ω,x) Φ( h )d τ + Ψ( lnr )d τ < . | | ≤ | | | | ∞ Due to linearity, CA (c1 + c2 lnr )h (r,ω,x) is τ-integrable. So the dominating function is integrable. | | Lemma 4.18. The uniform integrability assumption (3.0.1) of the theorem holds if: (1) c : esssup r k C var F , B◦ x,ar sup = rˆ>0,xˆ F ˆ− k rˆ (ˆ ˆ) < ∈ ∞ (µ-essential in x,ˆ Lebesgue in rˆ ) or (2) k = d (Minkowski content) or (3) k = d 1 (surface content). − Proof. We will reduce the first assertion to the preceding Lemma 4.17. Define xˆ := k ,var φ x, rˆ := r φ u . The proof of Lemma 4.14 (localizing and scaling π = ρk , ) ωÞn ωÞ0 n ωÞn ... ± shows| | | k var k var r C ψ F ,ψ B◦ x,ar rˆ C Fr , B◦ xˆ,ar csup − k ωßm,u r ωßm,u ( ) = − k ˆ ( ) | | ≤ whenever r ψ x R (x,0). This and then K 1 ψ x K imply for the pre- ω0ßm,u − ω0ßm,u ceding Lemma’s≤ h |(4.7.1), ≤ | ≤   M 1 1 X−  r  h (r,ω,x) sup csup 1(R(x,1),R(x,0)]   csup 1(R(x,1)/K ,R(x,0)K ] (r ). M >0 M   ≤ m=0 ψ x ≤ ω0ßm,u | The measure d r, ,x : r 11 r d r d ,x is finite, see Lemma 4.11 τ( ω ) = − (R(x,1)/K ,R(x,0)K ] ( ) νbi (ω ) (5). As a constant function, csup belongs to its Zygmund space. The other assertions reduce to the first. In case k = d (volume), the supremum condition is trivial. The case k = d 1 (surface area) is proved in [RZ10, Remark 3.2.2]. (Although stated only for self-similar− F , the proof works for general compact sets.) CURVATURES OF SELF-CONFORMAL SETS 32

5. APPENDIX:VECTOR TOEPLITZTHEOREM

d Lemma 5.1. Let αn ,α be ﬁnite, signed Borel measures on R , n N, such that the weak*-limit (point-wise∞ limit on continuous functions) ∈

wlimn αn = α →∞ ∞ exists. Let bn ,b be a bounded family of equicontinuous functions whose averages converge in supremum∞ norm 1 X lim bn = b . N N ∞ →∞ n N Then the following limit exists: ≤ 1 X lim αn (bn ) = α (b ). N N ∞ ∞ →∞ n N ≤ Proof. We will prove2 the assertion in only one dimension for simplicity and because product functions are dense. All measures shall be supported in x,y , neglecting tails as the family is tight. Due to the Banach-Steinhaus theorem, supn αn C x,y < . Since 0[ ] the quality of approximation is determined by the norm and the modulusk k of continuity∞ only ([Wer00, Thm. 1.2.10]), we can equiuniformly approximate the bn with Bernstein polynomials pt using a ﬁxed, ﬁnite set T of points,

X bn pt bn (t ) < ε. − t T C x,y ∈ [ ] By combining both estimates, we will be ﬁnished once we can show ! ! 1 X X X lim αn pt bn (t ) = α pt b (t ) . N N ∞ ∞ →∞ n N t T t T ≤ ∈ ∈ This reduces the assertion to the sequences of reals: consider a n := αn pt , bn (t ) for each point t T separately. There, it is a special case of Toeplitz’s theorem [Wer00, Thm. ∈ 4.4.6] for the resummation method 1 n N bn /N applied to a n . { ≤ } 6. APPENDIX:EXAMPLE

d The group of Möbius transformations of R is generated by similarities and the Möbius 2 d inversion x x/ x . The Möbius transformation φ specified by its pole p R 7→ | | d ∈ ∪ {∞} and the derivative r A (r > 0, A O (d )) at its fixed point f R is given by ∈ ∈ 2 φx = f + r A x f +O x f − −  2  p f (6.0.1) f r A id 2Proj p f  x p p f , = + − − 2 + p f p x − | − | − − − where Projy denotes the orthogonal projection onto y . The example in the introduction is generated by three Möbius maps. They agree with those of the classical Sierpinski triangle up to first order at their fixed points, except we reduce the contraction ratio to re-obtain the Open Set Condition. All three maps φi 5 share ri = 785/2000, Ai = id for i I = 1,2,3 . But f 1 = (0,0), p1 = 3, 2 and f 2 = (1,0), 1 p3 ∈ { } p2 = ( 3, 2) and f 3 = 2 , 2 , p3 = ( 2, 4) differ. − − − − 2 A direct proof is easier than checking the topological assumptions of [CP69]. CURVATURES OF SELF-CONFORMAL SETS 33

REFERENCES

[Ber03a] Andreas Bernig, On some aspects of curvature, 2003, (preprint) http://www.uni- frankfurt.de/fb/fb12/mathematik/an/bernig/bernig_survey.pdf. [Ber03b] , Variation of curvatures of subanalytic spaces and Schläﬂi-type formulas, Ann. Global Anal. Geom. 24 (2003), no. 1, 67–93. [BZ11] Tilman Johannes Bohl and Martina Zähle, Curvature-direction measures of self-similar sets, Arxiv 1111.4457. [CP69] R. H. Cox and R. E. Powell, Regularity of net summability transforms on certain linear topological spaces, Proc. Amer. Math. Soc. 21 (1969), 471–476. [CR09] Jean-Pierre Conze and Albert Raugi, On the ergodic decomposition for a cocycle, Colloq. Math. 117 (2009), no. 1, 121–156. [CSM06] David Cohen-Steiner and Jean-Marie Morvan, Second fundamental measure of geometric sets and local approximation of curvatures, J. Differential Geom. 74 (2006), no. 3, 363–394. [Dol02] Dmitry Dolgopyat, On mixing properties of compact group extensions of hyperbolic systems, Israel J. Math. 130 (2002), 157–205. [Doo94] J. L. Doob, Measure theory, Graduate Texts in Mathematics, vol. 143, Springer-Verlag, New York, 1994. [Fal95] K. J. Falconer, On the Minkowski measurability of fractals, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1115–1124. [Fed59] Herbert Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491. [FK11] Uta Freiberg and Sabrina Kombrink, Minkowski content and local Minkowski content for a class of self-conformal sets, Geometriae Dedicata (2011), 1–19, 10.1007/s10711-011-9661-5. [Fu85] Joseph Howland Guthrie Fu, Tubular neighborhoods in Euclidean spaces, Duke Math. J. 52 (1985), no. 4, 1025–1046. [Gat00] Dimitris Gatzouras, Lacunarity of self-similar and stochastically self-similar sets, Trans. Amer. Math. Soc. 352 (2000), no. 5, 1953–1983. [Geh92] F. W. Gehring, Topics in quasiconformal mappings, Quasiconformal space mappings, Lecture Notes in Math., vol. 1508, Springer, Berlin, 1992, pp. 20–38. [HLW04] Daniel Hug, Günter Last, and Wolfgang Weil, A local Steiner-type formula for general closed sets and applications, Math. Z. 246 (2004), no. 1-2, 237–272. [KK10] Marc Kesseböhmer and Sabrina Kombrink, Fractal curvature measures and Minkowski content for one-dimensional self-conformal sets, Advances of Mathematics (to appear) (2010), Arxiv 1012.5399. [Kom11] Sabrina Kombrink, Fractal curvature measures and minkowski content for limit sets of conformal function systems, Ph.D. thesis, Universität Bremen, 2011. [KP08] Steven G. Krantz and Harold R. Parks, Geometric integration theory, Cornerstones, Birkhäuser Boston Inc., Boston, MA, 2008. [LP93] Michel L. Lapidus and Carl Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3) 66 (1993), no. 1, 41–69. [LPW11] Michel L. Lapidus, Erin P.J. Pearse, and Steffen Winter, Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators, Adv. Math. 227 (2011), no. 4, 1349–1398. [MU96] R. Daniel Mauldin and Mariusz Urbanski,´ Dimensions and measures in inﬁnite iterated function systems, Proc. London Math. Soc. (3) 73 (1996), no. 1, 105–154. [MU03] , Graph directed Markov systems, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003, Geometry and dynamics of limit sets. [Pat97] Norbert Patzschke, Self-conformal multifractal measures, Adv. in Appl. Math. 19 (1997), no. 4, 486–513. [Pat04] , The tangent measure distribution of self-conformal fractals, Monatsh. Math. 142 (2004), no. 3, 243–266. [Pet89] Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1989, Corrected reprint of the 1983 original. [Pok11] D. Pokorny, On critical values of self-similar sets, Houston J. Math (to appear) (2011), Arxiv 1101.1219v3. [PP97] William Parry and Mark Pollicott, The Livšic cocycle equation for compact Lie group extensions of hyperbolic systems, J. London Math. Soc. (2) 56 (1997), no. 2, 405–416. CURVATURES OF SELF-CONFORMAL SETS 34

[PRSS01] Yuval Peres, Michał Rams, Károly Simon, and Boris Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proc. Amer. Math. Soc. 129 (2001), no. 9, 2689–2699 (electronic). [PU10] Feliks Przytycki and Mariusz Urbanski,´ Conformal fractals: ergodic theory methods, London Mathematical Society Lecture Note Series, vol. 371, Cambridge University Press, Cambridge, 2010. [Rau07] Albert Raugi, Mesures invariantes ergodiques pour des produits gauches, Bull. Soc. Math. France 135 (2007), no. 2, 247–258. [RR91] M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991. [RSM09] Jan Rataj, Evgeny Spodarev, and Daniel Meschenmoser, Approximations of the Wiener sausage and its curvature measures, Ann. Appl. Probab. 19 (2009), no. 5, 1840–1859. [RW10] Jan Rataj and Steffen Winter, On volume and surface area of parallel sets, Indiana Univ. Math. J. 59 (2010), no. 5, 1661–1685. [RW12] J. Rataj and S. Winter, Characterization of Minkowski measurability in terms of surface area, J. of Math. Analysis and Application (2012), Arxiv 1111.1825v2. [RZ01] J. Rataj and M. Zähle, Curvatures and currents for unions of sets with positive reach. II, Ann. Global Anal. Geom. 20 (2001), no. 1, 1–21. [RZ03] , Normal cycles of Lipschitz manifolds by approximation with parallel sets, Differential Geom. Appl. 19 (2003), no. 1, 113–126. [RZ10] , Curvature densities of self-similar sets, Indiana Univ. Math. J. (to appear) (2010), Arxiv 1009.6162. [Sta76] L. L. Stachó, On the volume function of parallel sets, Acta Sci. Math. (Szeged) 38 (1976), no. 3–4, 365–374. [STMK+11] G.E. Schröder-Turk, W. Mickel, S.C. Kapfer, M.A. Klatt, F.M. Schaller, M.J.F. Hoffmann, N. Klepp- mann, P.Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, Minkowski tensor shape analysis of cellular, granular and porous structures, Advanced Materials 23 (2011), no. 22-23, 2535–2553. [SW92] Rolf Schneider and Wolfgang Weil, Integralgeometrie, Teubner Skripten zur Mathematischen Stochastik. [Teubner Texts on Mathematical Stochastics], B. G. Teubner, Stuttgart, 1992. [Wer00] Dirk Werner, Funktionalanalysis, extended ed., Springer-Verlag, Berlin, 2000. [Win08] Steffen Winter, Curvature measures and fractals, Dissertationes Math. (Rozprawy Mat.) 453 (2008), 66. [Win11] , Curvature bounds for neighborhoods of self-similar sets, Comment. Math. Univ. Carolin. 52 (2011), no. 2, 205–226, Arxiv 1010.2032. [WZ12] Steffen Winter and Martina Zähle, Fractal curvature measures of self-similar sets, Adv. in Geom. (2012), Arxiv 1007.0696. [Zäh86] M. Zähle, Integral and current representation of Federer’s curvature measures, Arch. Math. (Basel) 46 (1986), no. 6, 557–567. [Zäh90] , Approximation and characterization of generalised Lipschitz-Killing curvatures, Ann. Global Anal. Geom. 8 (1990), no. 3, 249–260. [Zäh01] , The average density of self-conformal measures, J. London Math. Soc. (2) 63 (2001), no. 3, 721–734. [Zäh11] , Lipschitz-Killing curvatures of self-similar random fractals, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2663–2684. E-mail address: [email protected]

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