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Multifractal formalism for Benedicks–Carleson quadratic maps

Published online by Cambridge University Press:  11 March 2013

YONG MOO CHUNG  and
HIROKI TAKAHASI
YONG MOO CHUNG
Affiliation:
Department of Applied Mathematics, Hiroshima University, Higashi-Hiroshima 739-8527, Japan email chung@amath.hiroshima-u.ac.jp
HIROKI TAKAHASI
Affiliation:
Department of Electronic Science and Engineering, Kyoto University, Kyoto 606-8501, Japan email takahasi.hiroki.7r@kyoto-u.ac.jp
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Abstract

For a positive measure set of non-uniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given continuous function and consider the associated Birkhoff spectrum which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to ‘see’ sufficiently many points. To this end, we construct a family of towers. Using these towers we establish a large deviation principle of empirical distributions, with Lebesgue as a reference measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 4 , August 2014 , pp. 1116 - 1141
Copyright
Copyright ©2013 Cambridge University Press 

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