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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Variational principles and mixed multifractal spectra


Authors: L. Barreira and B. Saussol
Journal: Trans. Amer. Math. Soc. 353 (2001), 3919-3944
MSC (2000): Primary 37D35, 37C45.
DOI: https://doi.org/10.1090/S0002-9947-01-02844-6
Published electronically: June 6, 2001
MathSciNet review: 1837214
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Abstract:

We establish a ``conditional'' variational principle, which unifies and extends many results in the multifractal analysis of dynamical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new multifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy.

Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the entropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the ``mixed'' multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the ``non-mixed'' multifractal spectra.


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Additional Information

L. Barreira
Affiliation: Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
Email: luis.barreira@math.ist.utl.pt

B. Saussol
Affiliation: LAMFA / CNRS FRE 2270, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens, France
Email: benoit.saussol@mathinfo.u-picardie.fr

DOI: https://doi.org/10.1090/S0002-9947-01-02844-6
Keywords: Dimension spectrum, entropy spectrum, variational principle
Received by editor(s): April 17, 2000
Received by editor(s) in revised form: August 30, 2000
Published electronically: June 6, 2001
Additional Notes: L. Barreira was partially supported by FCT’s Funding Program and NATO grant CRG970161. B. Saussol was partially supported by FCT’s Funding Program and by the Center for Mathematical Analysis, Geometry, and Dynamical Systems.
Article copyright: © Copyright 2001 American Mathematical Society