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

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.

scholarly journals Large deviation principles for non-uniformly hyperbolic rational maps

2010 ◽  
Vol 31 (2) ◽  
pp. 321-349 ◽  
Author(s):  
HENRI COMMAN ◽  
JUAN RIVERA-LETELIER
Keyword(s):  
Large Deviation ◽  
Periodic Points ◽  
Rational Maps ◽  
Pressure Function ◽  
Hölder Continuous ◽  
Large Deviation Principles ◽  
Continuous Potential ◽  
Uniform Hyperbolicity ◽  
Level 2 ◽  
Unique Equilibrium State

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.

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

2014 ◽  
Vol 36 (1) ◽  
pp. 127-141 ◽  
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.

scholarly journals Multifractal formalism for Benedicks–Carleson quadratic maps

2013 ◽  
Vol 34 (4) ◽  
pp. 1116-1141 ◽  
Author(s):  
YONG MOO CHUNG ◽  
HIROKI TAKAHASI
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Probability Measures ◽  
Multifractal Formalism ◽  
Quadratic Maps ◽  
Empirical Distributions ◽  
Reference Measure ◽  
Time Averages

AbstractFor 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.

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

2015 ◽  
Vol 37 (1) ◽  
pp. 79-102 ◽  
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.

Large deviation principle for piecewise monotonic maps with density of periodic measures

2021 ◽  
pp. 1-12
Author(s):  
YONG MOO CHUNG ◽  
KENICHIRO YAMAMOTO
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Broad Class ◽  
Maximal Entropy ◽  
Deviation Principle ◽  
Piecewise Monotonic ◽  
Ergodic Measures ◽  
Level 2 ◽  
Markov Diagram ◽  
Measure Of Maximal Entropy

Abstract We show that a piecewise monotonic map with positive topological entropy satisfies the level-2 large deviation principle with respect to the unique measure of maximal entropy under the conditions that the corresponding Markov diagram is irreducible and that the periodic measures of the map are dense in the set of ergodic measures. This result can apply to a broad class of piecewise monotonic maps, such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces.

Thermodynamic formalism for a modified shift map

2014 ◽  
Vol 14 (02) ◽  
pp. 1350020 ◽  
Author(s):  
Stephen Muir ◽  
Mariusz Urbański
Keyword(s):  
A Priori ◽  
Transfer Operator ◽  
Thermodynamic Formalism ◽  
Lattice Gases ◽  
Hölder Continuous ◽  
Full Generality ◽  
Counting Measure ◽  
Shift Map ◽  
Continuous Potential ◽  
Shift Action

We introduce a transfer operator and use it to prove some theorems of a classical flavor from thermodynamic formalism (including existence and uniqueness of appropriately defined Gibbs states and equilibrium states for potential functions satisfying Dini's condition and stochastic laws for Hölder continuous potential and observable functions) in a novel setting: the "alphabet" E is a compact metric space equipped with an a priori probability measure ν and an endomorphism T. The "modified shift map" S is defined on the product space Eℕ by the rule (x1x2x3…) ↦ (T(x2)x3…). The greatest novelty is found in the variational principle, where a term must be added to the entropy to reflect the transformation of the first coordinate by T after shifting. Our motivation is that this system, in its full generality, cannot be treated by the existing methods of either rigorous statistical mechanics of lattice gases (where only the true shift action is used) or dynamical systems theory (where the a priori measure is always implicitly taken to be the counting measure).

Large deviation principle for the field in prequantum classical statistical field theory

2017 ◽  
Vol 20 (04) ◽  
pp. 1750025
Author(s):  
Andrei Khrennikov ◽  
Achref Majid
Keyword(s):  
Field Theory ◽  
Random Field ◽  
Field Model ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Background Field ◽  
Deviation Principle ◽  
Statistical Field ◽  
Statistical Field Theory

In this paper, we prove a large deviation principle for the background field in prequantum statistical field model. We show a number of examples by choosing a specific random field in our model.

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