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On a class of stable conditional measures

Published online by Cambridge University Press:  01 November 2010

EUGEN MIHAILESCU
EUGEN MIHAILESCU*
Affiliation:
Institute of Mathematics, ‘Simion Stoilow’ of the Romanian Academy, PO Box 1-764, RO-014700, Bucharest, Romania (email: Eugen.Mihailescu@imar.ro)
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Abstract

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The dynamics of endomorphisms (smooth non-invertible maps) presents many differences from that of diffeomorphisms or that of expanding maps; most methods from those cases do not work if the map has a basic set of saddle type with self-intersections. In this paper we study the conditional measures of a certain class of equilibrium measures, corresponding to a measurable partition subordinated to local stable manifolds. We show that these conditional measures are geometric probabilities on the local stable manifolds, thus answering in particular the questions related to the stable pointwise Hausdorff and box dimensions. These stable conditional measures are shown to be absolutely continuous if and only if the respective basic set is a non-invertible repeller. We find also invariant measures of maximal stable dimension, on folded basic sets. Examples are given, too, for such non-reversible systems.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 5 , October 2011, pp. 1499 - 1515
Copyright
Copyright © Cambridge University Press 2010

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