Expanding maps on sets which are almost invariant. Decay and chaos

Abstract:

Let $A \subset {R^n}$ be a bounded open set with finitely many connected components ${A_j}$ and let $T: \overline A \to {R^n}$ be a smooth map with $A \subset T\left ( A \right )$. Assume that for each ${A_j}$, $A \subset {T^m}\left ( {{A_j}} \right )$ for all m sufficiently large. We assume that T is “expansive", but we do not assume that $T\left ( A \right ) = A$. Hence for $x \in A, {T^i} \left ( x \right )$ may escape A as i increases. Let ${\mu _0}$ be a smooth measure on A (with ${\operatorname {inf} _A} {{d{\mu _0}} \left / {\vphantom {{d{\mu _0}} {dx}}} \right . {dx}} > 0$) and let $x \in A$ be chosen at random (using ${\mu _0}$). Since T is “expansive” we may expect ${T^i}\left ( x \right )$ to oscillate chaotically on A for a certain time and eventually escape A. For each measurable set $E \subset A$ define ${\mu _m}\left ( E \right )$ to be the conditional probability that ${T^m}\left ( x \right ) \in E$ given that $x,T\left ( x \right ),\ldots ,{T^m}\left ( x \right )$ are in A. We show that ${\mu _m}$ converges to a smooth measure $\mu$ which is independent of the choice of ${\mu _0}$. One dimensional examples are stressed.