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The thermodynamic approach to multifractal analysis

Published online by Cambridge University Press:  04 August 2014

VAUGHN CLIMENHAGA
VAUGHN CLIMENHAGA*
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA email climenha@math.uh.edu
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Abstract

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Most results in multifractal analysis are obtained using either a thermodynamic approach based on the existence and uniqueness of equilibrium states or an orbit-gluing approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase transitions, and obtain new results for a number of examples that have already been studied using the orbit-gluing approach. We compare the thermodynamic and orbit-gluing approaches and give a survey of many of the multifractal results in the literature.

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Type
Survey
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 5 , October 2014, pp. 1409 - 1450
Copyright
© Cambridge University Press, 2014 

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