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Dimensions of compact invariant sets of some expanding maps

Published online by Cambridge University Press:  01 February 2009

YUKI YAYAMA
YUKI YAYAMA*
Affiliation:
Centro de Modelamiento Matemático, Universidad de Chile, Avenue Blanco Encalada, 2120 Piso 7, Santiago, Chile (email: yyayama@dim.uchile.cl)
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Abstract

We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding non-conformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a ‘general Sierpiński carpet’ was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by using compensation functions to study a general Sierpiński carpet represented by a shift of finite type. We give some conditions under which a general Sierpiński carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measure.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 1 , February 2009 , pp. 281 - 315
Copyright
Copyright © 2008 Cambridge University Press

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