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 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 Relative multifractal box-dimensions

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2841-2859 ◽  
Author(s):  
Najmeddine Attia ◽  
Bilel Selmi
Keyword(s):  
Hausdorff Dimension ◽  
Multifractal Spectrum ◽  
Packing Dimension ◽  
Probability Measures ◽  
Multifractal Formalism ◽  
The Relationship ◽  
Box Dimensions

Given two probability measures ? and ? on Rn. We define the upper and lower relative multifractal box-dimensions of the measure ? with respect to the measure ? and investigate the relationship between the multifractal box-dimensions and the relative multifractal Hausdorff dimension, the relative multifractal pre-packing dimension. We also, calculate the relative multifractal spectrum and establish the validity of multifractal formalism. As an application, we study the behavior of projections of measures obeying to the relative multifractal formalism.

scholarly journals Lyapunov spectrum for Hénon-like maps at the first bifurcation

2016 ◽  
Vol 38 (3) ◽  
pp. 1168-1200
Author(s):  
HIROKI TAKAHASI
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Level Sets ◽  
Multifractal Analysis ◽  
Lyapunov Spectrum ◽  
Probability Measures ◽  
Limit Set ◽  
Uniform Hyperbolicity ◽  
Partial Description ◽  
Set Of Points

For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.

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.

scholarly journals The Hausdorff dimension of level sets described by Erdös–Rényi average

2018 ◽  
Vol 458 (1) ◽  
pp. 464-480
Author(s):  
Haibo Chen ◽  
Daoxin Ding ◽  
Xinghuo Long
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets

Exceptional sets related to the largest digits in Lüroth expansions

2021 ◽  
pp. 1-15
Author(s):  
Shuyi Lin ◽  
Jinjun Li ◽  
Manli Lou
Keyword(s):  
Number Theory ◽  
Hausdorff Dimension ◽  
Level Sets ◽  
Disjoint Union ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Lüroth Expansion

Let [Formula: see text] denote the largest digit of the first [Formula: see text] terms in the Lüroth expansion of [Formula: see text]. Shen, Yu and Zhou, A note on the largest digits in Luroth expansion, Int. J. Number Theory 10 (2014) 1015–1023 considered the level sets [Formula: see text] and proved that each [Formula: see text] has full Hausdorff dimension. In this paper, we investigate the Hausdorff dimension of the following refined exceptional set: [Formula: see text] and show that [Formula: see text] has full Hausdorff dimension for each pair [Formula: see text] with [Formula: see text]. Combining the two results, [Formula: see text] can be decomposed into the disjoint union of uncountably many sets with full Hausdorff dimension.

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