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Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 37 / Issue 1 / February 2017
- Published online by Cambridge University Press:
- 12 October 2015, pp. 79-102
- Print publication:
- February 2017
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- Article
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In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if
$f$ has an expanding repeller and 
$\unicode[STIX]{x1D719}$ is a Hölder continuous potential, we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval 
$I\subset \mathbb{R}$ can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.
