Escape rates for Gibbs measures

Abstract

Abstract. In this paper we study the asymptotic behaviour of the escape rate of a Gibbs measure supported on a conformal repeller through a small hole. There are additional applications to the convergence of the Hausdorff dimension of the survivor set. IntroductionGiven any transformation T : X → X preserving an ergodic probability measure µ and any Borel set A ⊂ X , the escape rate quantifies the asymptotic behaviour of the measure of the set of points x ∈ X for which none of the first n terms in the orbit intersects U . Bunimovich and Yurchenko [7] considered the fundamental case of the doubling map and Haar measure, and where U is a dyadic interval. Subsequently, Keller and Liverani [16] proved a more general perturbation result which was then used to show, amongst other things, that a similar formula holds in the case that T is an expanding interval map and µ the absolutely continuous invariant probability measure. Other papers regarding this topic include [1,6,9,19] and references therein.In this paper, we prove analogous results in the more general setting of Gibbs measures supported on conformal repellers. Much of the analysis is undertaken in the setting of subshifts of finite type; this not only allows us to prove similar results for a broad class of maps which can be modelled symbolically but also improve on the work of Lind [18] who considered the convergence of topological entropy for a topologically mixing subshift.Another interesting aspect of our analysis is the connection with the work of Hirata [12] on the exponential law for first return times for Axiom A diffeomorphisms. Some of the ingredients in our approach were suggested by Hirata's paper, although we had to significantly modify the actual details.Let M be a Riemannian manifold and f : M → M a C 1 -map. Let J be a compact subset of M such that f (J ) = J . We say that the pair (J, f ) is a conformal repeller if: (1) f | J is a conformal map; (2) there exist c > 0 and λ > 1 such that d f n x v ≥ cλ n v for all x ∈ J , v ∈ T x M, and n ≥ 1; (3) f is topologically mixing on J ;

Extremes and Recurrence in Dynamical Systems

“…This means that Theorem 5.4.1 holds, but for two-sided kcylinders defined at points in Λ rather than balls. The approximation results of balls by cylinders, performed in [147,146] means that the result also extends to balls. The application given in [147] is to Axiom A diffeomorphisms, e.g., the so-called Arnold cat map given by x n+1 y n+1 = 2 1 1 1 x n y n on the 2-torus, since it is well-known that these have the requisite Markov partition, see, e.g.…”

“…We will formalise these results in a theorem using the more complete picture, given by the full treatment made in [146], which builds up on the work developed in [147] and also, to some extent, in [150]. The method there was a transfer operator approach.…”

Dimension of generic self-affine sets with holes

“…In many systems the escape rate can be described in terms of the μ-measure of the hole. In particular, often the escape rate and measure of the hole can also be used to quantify the asymptotic rate of decrease of the dimension deficit; that is, speed at which the dimension of the system with a hole approaches the dimension of the full system [4,8,12,16]. Recently, some interest has arisen in studying classical dynamical problems on self-affine fractal sets, under the dynamics that naturally arises from the definition of the set via an iterated function system [2,11,17] (for definitions, see Sect.…”

Behaviour of the escape rate function in hyperbolic dynamical systems

“…Also, although we usually suppress the parameters, ρ depends on both the hole, H, and on the measure, µ. In the present work, µ will always be either Lebesgue measure or the Sinai, Ruelle, Bowen (SRB) measure for f , and we study the behavior of the escape rate as a function of the hole.Many works on systems with holes have focused on the existence of quasi-invariant measures with physical properties by assuming either the existence of a finite Markov partition The derivative of the escape rate in the zero-hole limit has also received attention recently [BY,KL2,FP].In this paper we consider both small and large holes and make no Markovian assumptions on the dynamics of the open systems. Let H t be a 1-parameter family of holes varying continuously with t. Let ρ(t) = ρ(H t , µ), when defined.…”

“…The derivative of the escape rate in the zero-hole limit has also received attention recently [BY,KL2,FP].…”