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Multifractal analysis of weak Gibbs measures and phase transition—application to some Bernoulli convolutions

Published online by Cambridge University Press:  02 December 2003

DE-JUN FENG  and
ERIC OLIVIER
DE-JUN FENG
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People's Republic of China (e-mail: dfeng@math.tsinghua.edu.cn) and Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
ERIC OLIVIER
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
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Abstract

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For a given expanding d-fold covering transformation of the one-dimensional torus, the notion of weak Gibbs measure is defined by a natural generalization of the classical Gibbs property. For these measures, we prove that the singularity spectrum and the $L^q$-spectrum form a Legendre transform pair. The main difficulty comes from the possible existence of first-order phase transition points, that is, points where the $L^q$-spectrum is not differentiable. We give examples of weak Gibbs measure with phase transition, including the so-called Erdös measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 23 , Issue 6 , December 2003 , pp. 1751 - 1784
Copyright
2003 Cambridge University Press