Markov extensions and natural conditionally invariant measures for dynamical systems with holes.

摘要

Consider a piecewise differentiable map T of a compact Riemannian manifold M. Let H be an open set in M. We call H the hole and keep track of the iterates of a point as long as they do not enter H. Once a point enters H, it is absorbed and does not return. We study the statistical properties of the escape dynamics of such a system and are interested in the existence and properties of absolutely continuous conditionally invariant measures (abbreviated a.c.c.i.m.).; We first show that a dynamical system with arbitrarily small holes may have many a.c.c.i.m.s which share the same support even when the corresponding closed system has been shown to have a unique absolutely continuous invariant measure. Many of these measures may not reflect the escape dynamics with respect to the reference measure on M. It then becomes essential to establish properties of a conditionally invariant measure, rather than merely prove its existence. We propose some properties that a natural conditionally invariant measure should have in order to shed light on the statistical properties of the system.; We introduce to the study of dynamical systems with holes the method of Markov extensions, due to L.-S. Young, which has been used successfully to study a wide variety of dynamical systems without holes. The basic idea of this method is to construct an associated dynamical system with simpler properties than the original, study the properties of this system and then pass the results back to the original system. This method does not require any Markov structure in the original dynamical system and is flexible enough to apply to nonuniformly hyperbolic systems. We first prove results for the of dynamical systems with holes: expanding maps of the interval and certain parameter values of the logistic family fa(x) = 1 - ax2 on [-1, 1]. This method yields the existence of an a.c.c.i.m. with many of the properties proposed for a natural conditionally invariant measure and has the potential to be extended to other classes of dynamical systems with holes.