Rates of Recurrence for Zq and Rq Extensions of Subshifts of Finite Type

In this note we give new asymptotic formulae for certain countingfunctions associated to the periodic behaviour of Z^q and R^qextensions of subshifts of finite type. In the case of the Z^qextensions, these strengthen previous estimates of Marcus andTuncel [9]. For both types of extension, our results complementthe central limit type results of Lalley [6]. Our proof requiresthe application of ideas from thermodynamic formalism. Whilstdeveloping this approach, in Section 2, we take the opportunityto present a counter-example to a related conjecture of Coelho-Filho[2].     

The dimensions of some self-affine limit sets in the plane and hyperbolic sets

In this article we compute the Hausdorff dimension and box dimension (or capacity) of a dynamically constructed model similarity process in the plane with two distinct contraction coefficients. These examples are natural generalizations to the plane of the simple Markov map constructions for Cantor sets on the line. Some related problems have been studied by different authors; however, those results are directed toward generic results in quite general situations. This paper concentrates on computing explicit formulas in as many specific cases as possible. The techniques of previous authors and ours are correspondingly very different. In our calculations, delicate number-theoretic properties of the contraction coefficients arise. Finally, we utilize the results for the model problem to compute the dimensions of some affine horseshoes in ⁿ , and we observe that the dimensions do not always coincide and their coincidence depends on delicate number-theoretic properties of the Lyapunov exponents.     

Statistical Properties of Maps¶with Indifferent Periodic Points

In this note we present an axiomatic approach to the decay of correlations for maps of arbitrary dimension with indifferent periodic points. As applications, we apply our results to the well-known Manneville–Pomeau equation and the inhomogeneous diophantine approximation algorithm. Received: 5 November 1999 / Accepted: 11 October 2000     

Growth series for the commutator subgroup

In this paper we study growth series associated to commutator subgroups of surface groups. We conclude that these growth series are not rational.

     

C k-rigidity for hyperbolic flows II

In this note we prove the following result: Any conjugating homeomorphism between two geodesic flows for compact negatively curved compactC ^∞ surfaces is necessarilyC ^∞. This extends a result of Feldman and Ornstein. We also discuss some related results for hyperbolic flows and diffeomorphisms.     

Exponential mixing for the geodesic flow on hyperbolic three-manifolds

We give a short and direct proof of exponential mixing of geodesic flows on compact hyperbolic three-manifolds with respect to the Liouville measure. This complements earlier results of Collet-Epstein-Gallovotti, Moore, and Ratner for hyperbolic surfaces. Furthermore, since the analysis is even easier in three dimensions than in two dimensions (because of the absence of discrete series and the simplicity of the zonal spherical functions in this case), this apparently gives the simplest example of a flow with exponential mixing.     

Livsic theorems for connected Lie groups

Let be a hyperbolic diffeomorphism on a basic set and let be a connected Lie group. Let be Hölder. Assuming that satisfies a natural partial hyperbolicity assumption, we show that if is a measurable solution to a.e., then must in fact be Hölder. Under an additional centre bunching condition on , we show that if assigns `weight' equal to the identity to each periodic orbit of , then for some Hölder . These results extend well-known theorems due to Livsic when is compact or abelian.

     

How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism

A popular wavelet reference [W] states that “in theoretical and practical studies, the notion of (wavelet) regularity has been increasing in importance.” Not surprisingly, the study of wavelet regularity is currently a major topic of investigation. Smoother wavelets provide sharper frequency resolution of functions. Also, the iterative algorithms to construct wavelets converge faster for smoother wavelets. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of wavelets and to devise a faster algorithm for estimating regularity. The thermodynamic approach works equally well for compactly supported and non-compactly supported wavelets, and also applies to non-analytic wavelet filters. We present an algorithm for computing the Sobolev regularity of wavelets and prove that it converges with super-exponential speed. As an application we construct new examples of wavelets that are smoother than the Daubechies wavelets and have the same support. We establish smooth dependence of the regularity for wavelet families, and we derive a variational formula for the regularity. We also show a general relation between the asymptotic regularity of wavelet families and maximal measures for the doubling map. Finally, we describe how these results generalize to higher dimensional wavelets. The work of the second author was partially supported by a National Science Foundation grant DMS-0355180.