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

Abstract

Let A ⊂ R n A\, \subset \,{R^n} be a bounded open set with finitely many connected components A j {A_j} and let T : A ¯ → R n T:\,\overline A \to \,{R^n} be a smooth map with A ⊂ T ( A ) A\,\, \subset \,\,T\left ( A \right ) . Assume that for each A j {A_j} , A ⊂ T m ( 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 ( A ) = A T\left ( A \right ) = A . Hence for x ∈ A , T i ( x ) x\, \in \,A,\,{T^i}\,\left ( x \right ) may escape A as i increases. Let μ 0 {\mu _0} be a smooth measure on A (with inf A d μ 0 / d μ 0 d x d x > 0 {\operatorname {inf} _A}\,{{d{\mu _0}} \left / {\vphantom {{d{\mu _0}} {dx}}} \right . {dx}}\, > \,0 ) and let x ∈ A x\, \in \,A be chosen at random (using μ 0 {\mu _0} ). Since T is “expansive” we may expect T i ( x ) {T^i}\left ( x \right )\, to oscillate chaotically on A for a certain time and eventually escape A . For each measurable set E ⊂ A E\, \subset \,A define μ m ( E ) {\mu _m}\left ( E \right ) to be the conditional probability that T m ( x ) ∈ E {T^m}\left ( x \right ) \in \,E given that x , T ( x ) , … , T m ( x ) x,T\left ( x \right ),\ldots ,{T^m}\left ( x \right ) are in A . We show that μ m {\mu _m} converges to a smooth measure μ \mu which is independent of the choice of μ 0 {\mu _0} . One dimensional examples are stressed.