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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Giulio Pianigiani and James A. Yorke
Journal: Trans. Amer. Math. Soc. 252 (1979), 351-366
MSC: Primary 58F11; Secondary 28D15, 34C35, 58F12
MathSciNet review: 534126
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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}} \mathord{\left/ {\vphantom {{d{\mu _0}} {dx}}} \right. \kern-\nulldelimiterspace} {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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0534126-2
PII: S 0002-9947(1979)0534126-2
Keywords: Frobenius-Perron operator, invariant measures, expanding maps, chaos
Article copyright: © Copyright 1979 American Mathematical Society