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Invariant densities for piecewise linear maps of the unit interval

Published online by Cambridge University Press:  12 January 2009

PAWEŁ GÓRA
PAWEŁ GÓRA*
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montreal, Quebec, Canada H3G 1M8 (email: pgora@mathstat.concordia.ca)
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Abstract

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We find an explicit formula for the invariant density h of an arbitrary eventually expanding piecewise linear map τ of an interval [0,1]. We do not assume that the slopes of the branches are the same and we allow arbitrary number of shorter branches touching zero or touching one or hanging in between. The construction involves the matrix S which is defined in a way somewhat similar to the definition of the kneading matrix of a continuous piecewise monotonic map. Under some additional assumptions, we prove that if 1 is not an eigenvalue of S, then the dynamical system (τ,h⋅m) is ergodic with full support.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009, pp. 1549 - 1583
Copyright
Copyright © Cambridge University Press 2008

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