scholarly journals The thermodynamic approach to multifractal analysis

2014 ◽  
Vol 34 (5) ◽  
pp. 1409-1450 ◽  
Author(s):  
VAUGHN CLIMENHAGA
Keyword(s):  
Existence And Uniqueness ◽  
General Framework ◽  
Multifractal Analysis ◽  
Broad Class ◽  
Hyperbolic Systems ◽  
Thermodynamic Approach ◽  
Property A ◽  
Multifractal Formalism ◽  
Uniformly Hyperbolic Systems ◽  
Uniqueness Of Equilibrium

AbstractMost results in multifractal analysis are obtained using either a thermodynamic approach based on the existence and uniqueness of equilibrium states or an orbit-gluing approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase transitions, and obtain new results for a number of examples that have already been studied using the orbit-gluing approach. We compare the thermodynamic and orbit-gluing approaches and give a survey of many of the multifractal results in the literature.

scholarly journals Multifractal analysis of weak Gibbs measures for non-uniformly expanding C1 maps

2010 ◽  
Vol 31 (1) ◽  
pp. 143-164 ◽  
Author(s):  
THOMAS JORDAN ◽  
MICHAŁ RAMS
Keyword(s):  
Gibbs Measure ◽  
Hyperbolic System ◽  
Multifractal Analysis ◽  
Invariant Measures ◽  
Hyperbolic Systems ◽  
Gibbs Measures ◽  
Equilibrium States ◽  
Local Dimension ◽  
Uniformly Hyperbolic Systems ◽  
Hyperbolic Sets

AbstractWe will consider the local dimension spectrum of a weak Gibbs measure on a C1 non-uniformly hyperbolic system of Manneville–Pomeau type. We will present the spectrum in three ways: using invariant measures, ergodic invariant measures supported on hyperbolic sets and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.

scholarly journals On equilibrium states for partially hyperbolic horseshoes

2016 ◽  
Vol 38 (1) ◽  
pp. 301-335 ◽  
Author(s):  
I. RIOS ◽  
J. SIQUEIRA
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Systems ◽  
Three Dimensional ◽  
Small Variation ◽  
Equilibrium States ◽  
Dimensional System ◽  
Hölder Continuous ◽  
The Family ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We prove the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced by Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys.29 (2009), 433–474]. For the original three-dimensional system we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.

scholarly journals Multifractal analysis of ergodic averages in some non-uniformly hyperbolic systems

2015 ◽  
Vol 36 (7) ◽  
pp. 2334-2350 ◽  
Author(s):  
ZHENG YIN ◽  
ERCAI CHEN ◽  
XIAOYAO ZHOU
Keyword(s):  
Multifractal Analysis ◽  
Hyperbolic Systems ◽  
Ergodic Averages ◽  
Local Diffeomorphisms ◽  
Uniformly Hyperbolic Systems

This article is devoted to the study of the multifractal analysis of ergodic averages in some non-uniformly hyperbolic systems. In particular, our results hold for the robust classes of multidimensional non-uniformly expanding local diffeomorphisms and Viana maps.

scholarly journals Multifractal analysis of non-uniformly hyperbolic systems

2010 ◽  
Vol 177 (1) ◽  
pp. 125-144 ◽  
Author(s):  
Anders Johansson ◽  
Thomas M. Jordan ◽  
Anders Öberg ◽  
Mark Pollicott
Keyword(s):  
Multifractal Analysis ◽  
Hyperbolic Systems ◽  
Uniformly Hyperbolic Systems

ALMOST ADDITIVE THERMODYNAMIC FORMALISM: SOME RECENT DEVELOPMENTS

2010 ◽  
Vol 22 (10) ◽  
pp. 1147-1179 ◽  
Author(s):  
LUIS BARREIRA
Keyword(s):  
Variational Principle ◽  
Existence And Uniqueness ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Topological Pressure ◽  
Present Stage ◽  
The Past ◽  
Recent Developments ◽  
General Sequences ◽  
Uniqueness Of Equilibrium

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

scholarly journals Quasi-shadowing for partially hyperbolic diffeomorphisms

2014 ◽  
Vol 35 (2) ◽  
pp. 412-430 ◽  
Author(s):  
HUYI HU ◽  
YUNHUA ZHOU ◽  
YUJUN ZHU
Keyword(s):  
Spectral Decomposition ◽  
Hyperbolic Systems ◽  
Decomposition Theorem ◽  
Closing Lemma ◽  
Shadowing Property ◽  
Hyperbolic Diffeomorphisms ◽  
Uniformly Hyperbolic Systems ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

AbstractA partially hyperbolic diffeomorphism $f$ has the quasi-shadowing property if for any pseudo orbit $\{x_{k}\}_{k\in \mathbb{Z}}$, there is a sequence of points $\{y_{k}\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_{k})$ by a motion ${\it\tau}$ along the center direction. We show that any partially hyperbolic diffeomorphism has the quasi-shadowing property, and if $f$ has a $C^{1}$ center foliation then we can require ${\it\tau}$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^{0}$-perturbation. When $f$ has a uniformly compact $C^{1}$ center foliation, we also give partially hyperbolic diffeomorphism versions of some theorems which hold for uniformly hyperbolic systems, such as the Anosov closing lemma, the cloud lemma and the spectral decomposition theorem.

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