Equilibrium states, pressure and escape for multimodal maps with holes

Abstract

For a class of non-uniformly hyperbolic interval maps, we study rates of escape with respect to conformal measures associated with a family of geometric potentials. We establish the existence of physically relevant conditionally invariant measures and equilibrium states and prove a relation between the rate of escape and pressure with respect to these potentials. As a consequence, we obtain a Bowen formula: we express the Hausdorff dimension of the set of points which never exit through the hole in terms of the relevant pressure function. Finally, we obtain an expression for the derivative of the escape rate in the zero-hole limit.

On transfer operators and maps with random holes

“…From applications point of view, one can use our results to study statistical properties and dimension theory of orbits that do not visit other ergodic components in a random metastable system, such as the ones studied in [5], and to provide insight to geophysical models that study regions of the ocean with slow or poor mixing properties [9]. Mathematically, previous results on the Hausdorff dimension of the survival set were obtained in [19,15,10,25,28] for maps with deterministic holes. In our random setting, an interesting feature of our current work is that it uses tools from deterministic and closed systems to obtain results in random open systems: the transfer operator associated with our random open system can be reduced to a transfer operator associated with the closed deterministic system.…”

“…a cylinder of length 2. Now the set of symbolic sequences x ∈ {0, 1} N that do not contain any two a priori fixed words out of [00], [01], [10], [11] as a sub-word contains the sequences 0 ∞ , 1 ∞ or (01) ∞ .…”

Asymptotic escape rates and limiting distributions for multimodal maps

“…For t ∈ (t − , t + ), let s t := t + pt λ(µt) ∈ (0, 1] denote the local scaling exponent for m tφ−pt , see [DT1,Lemma 9.5]. Define D n (c) = |Df n (f (c))| for each c ∈ Crit .…”

“…Such results have been obtained primarily for uniformly hyperbolic systems, beginning with expanding maps [PY, CMS, LM], Anosov diffeomorphisms [CM, CMT], finite [FP] and countable [DIMMY] state topological Markov chains, and dispersing billiards [DWY,D2]. Their extension to non-uniformly hyperbolic systems has been primarily restricted to unimodal and multimodal interval maps [BDM,DT1,PU] and intermittent maps [DT2].…”

“…As such, the present paper represents a significant simplification and extension of results available in the context of non-uniformly hyperbolic open systems. Previous works in the setting of unimodal maps with holes have required strong conditions both on the map (Misiurewicz maps in [D1]; a Benedicks-Carleson condition in [BDM,DT1]; a topologically tame condition in [PU]) and on the placement of the hole (slow approach to (see [BDM,DT1]) or complete avoidance of (see [PU]) the post-critical set by the boundary or centre of the hole).…”