Large deviation principles for non-uniformly hyperbolic rational maps

AbstractWe show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called ‘Topological Collet–Eckmann’. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that each Hölder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer.

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Large deviation principles of one-dimensional maps for Hölder continuous potentials

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.55 ◽

2014 ◽

Vol 36 (1) ◽

pp. 127-141 ◽

Cited By ~ 2

Author(s):

HUAIBIN LI

Keyword(s):

Large Deviation Principle ◽

Large Deviation ◽

Weak Form ◽

Periodic Points ◽

Hölder Continuous ◽

One Dimensional ◽

Deviation Principle ◽

Large Deviation Principles ◽

One Dimensional Maps ◽

Level 2

We show some level-2 large deviation principles for real and complex one-dimensional maps satisfying a weak form of hyperbolicity. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages.

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Unique equilibrium states, large deviations and Lyapunov spectra for the Katok map

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.26 ◽

2020 ◽

pp. 1-38

Author(s):

TIANYU WANG

Keyword(s):

Equilibrium State ◽

Level Sets ◽

Additional Condition ◽

Large Deviation Principle ◽

Large Deviation ◽

Thermodynamic Formalism ◽

Hölder Continuous ◽

Lyapunov Spectra ◽

Continuous Potential ◽

Level 2

We study the thermodynamic formalism of a $C^{\infty }$ non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a Hölder continuous potential with one additional condition, or geometric $t$ -potential $\unicode[STIX]{x1D711}_{t}$ with $t<1$ , the equilibrium state exists and is unique. We derive the level-2 large deviation principle for the equilibrium state of $\unicode[STIX]{x1D711}_{t}$ . We study the multifractal spectra of the Katok map for the entropy and dimension of level sets of Lyapunov exponents.

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Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.46 ◽

2015 ◽

Vol 37 (1) ◽

pp. 79-102 ◽

Cited By ~ 9

Author(s):

THIAGO BOMFIM ◽

PAULO VARANDAS

Keyword(s):

Large Deviations ◽

Gibbs Measures ◽

Rate Function ◽

Topological Pressure ◽

Hölder Continuous ◽

Continuous Potential ◽

Hyperbolic Flows ◽

Time Averages ◽

Set Of Points ◽

Level 2

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.

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