Markov Extensions for Dynamical Systems with Holes: An Application to Expanding Maps of the Interval

We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions on the dynamics of the tower which ensure existence and uniqueness in a class of Holder continuous densities. We then use these results to study the existence and properties of a.c.c.i.m. for expanding maps of the interval with holes. We obtain the convergence of the a.c.c.i.m. to the SRB measure of the corresponding closed system as the measure of the hole shrinks to zero.Comment: 32 pages. New version contains minor revisions, primarily to clarify introductory Section

Equilibrium states, pressure and escape for multimodal maps with holes

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.Comment: Minor edits. To appear in Israel J. Mat

Existence and convergence properties of physical measures for certain dynamical systems with holes

We study two classes of dynamical systems with holes: expanding maps of the interval and Collet-Eckmann maps with singularities. In both cases, we prove that there is a natural absolutely continuous conditionally invariant measure $\mu$ (a.c.c.i.m.) with the physical property that strictly positive H\"{o}lder continuous functions converge to the density of $\mu$ under the renormalized dynamics of the system. In addition, we construct an invariant measure $\nu$, supported on the Cantor set of points that never escape from the system, that is ergodic and enjoys exponential decay of correlations for H\"{o}lder observables. We show that $\nu$ satisfies an equilibrium principle which implies that the escape rate formula, familiar to the thermodynamic formalism, holds outside the usual setting. In particular, it holds for Collet-Eckmann maps with holes, which are not uniformly hyperbolic and do not admit a finite Markov partition. We use a general framework of Young towers with holes and first prove results about the \accim and the invariant measure on the tower. Then we show how to transfer results to the original dynamical system. This approach can be expected to generalize to other dynamical systems than the two above classes.Comment: 44 pages. Major addition: this paper now treats Collet-Eckmann maps with singularitie

Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas

In this paper, we show how the Gordin martingale approximation method fits into the anisotropic Banach space framework. In particular, for the time-one map of a finite horizon planar periodic Lorentz gas, we prove that Holder observables satisfy statistical limit laws such as the central limit theorem and associated invariance principles.Comment: Final version, to appear in Nonlinearity. Corrected some minor typos from previous versio

Functional Norms for Young Towers

We introduce functional norms for hyperbolic Young towers which allow us to directly study the transfer operator on the full tower. By eliminating the need for secondary expanding towers commonly employed in this context, this approach simplifies and expands the analysis of this class of Markov extensions and the underlying systems for which they are constructed. As an example, we prove large-deviation estimates with a uniform rate function for a large class of non-invariant measures and show how to translate these to the underlying system