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Basins of measures on inverse limit spaces for the induced homeomorphism

Published online by Cambridge University Press:  13 October 2009

JUDY KENNEDY ,
BRIAN E. RAINES  and
DAVID R. STOCKMAN
JUDY KENNEDY
Affiliation:
Department of Mathematics, Lamar University, Beaumont, TX 77710, USA (email: [email protected])
BRIAN E. RAINES
Affiliation:
Department of Mathematics, Baylor University, Waco, TX 76798, USA (email: [email protected])
DAVID R. STOCKMAN
Affiliation:
Department of Economics, University of Delaware, Newark, DE 19716, USA (email: [email protected])
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Abstract

Let f:X→X be continuous and onto, where X is a compact metric space. Let be the inverse limit and F:Y →Y the induced homeomorphism. Suppose that μ is an f-invariant measure, and let m be the measure induced on Y by (μ,μ,…). We show that B is a basin of μ if and only if π−11(B) is a basin of m. From this it follows that if μ is an SRB measure for f on X, then the induced measure m on Y is an inverse-limit SRB measure for F. Conversely, if m is an inverse-limit SRB measure for F on Y, then the induced measure μ on X is an SRB measure for f.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 1119 - 1130
Copyright
Copyright © Cambridge University Press 2009

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