scholarly journals Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Suzete Maria Afonso ◽  
Vanessa Ramos ◽  
Jaqueline Siqueira
Keyword(s):  
Hyperbolic Systems ◽  
Statistical Properties ◽  
Equilibrium States ◽  
Uniformly Hyperbolic Systems

scholarly journals Equilibrium stability for non-uniformly hyperbolic systems

2018 ◽  
Vol 39 (10) ◽  
pp. 2619-2642 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
VANESSA RAMOS ◽  
JAQUELINE SIQUEIRA
Keyword(s):  
Hyperbolic Systems ◽  
Equilibrium States ◽  
Topological Pressure ◽  
Equilibrium Stability ◽  
Hölder Continuous ◽  
Skew Products ◽  
Uniformly Hyperbolic Systems ◽  
Wide Family

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally, we show that these equilibrium states vary continuously in the $\text{weak}^{\ast }$ topology within such systems.

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.

Weak Gibbs measures for certain non-hyperbolic systems

2000 ◽  
Vol 20 (5) ◽  
pp. 1495-1518 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Bounded Variation ◽  
Invariant Measures ◽  
Hyperbolic Systems ◽  
Gibbs Measures ◽  
Periodic Points ◽  
Statistical Properties ◽  
Equilibrium States ◽  
Pointwise Dimension ◽  
Absolutely Continuous ◽  
Higher Dimensional

We study a weak Gibbs property of equilibrium states for potentials of weak bounded variation and for maps admitting indifferent periodic points. We further establish statistical properties of the weak Gibbs measures and bounds of their pointwise dimension. We apply our results to higher-dimensional maps (which are not necessarily conformal) with indifferent periodic points and show that their absolutely continuous finite invariant measures are weak Gibbs measures.

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.

scholarly journals Symbolic dynamics for non-uniformly hyperbolic systems

2020 ◽  
pp. 1-68
Author(s):  
YURI LIMA
Keyword(s):  
Symbolic Dynamics ◽  
Hyperbolic Systems ◽  
Markov Partition ◽  
Equilibrium Measures ◽  
Markov Partitions ◽  
Recent Advances ◽  
Closed Orbits ◽  
Symbolic Extension ◽  
Uniformly Hyperbolic Systems

Abstract This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

On the convergence to equilibrium states for certain non-hyperbolic systems

1997 ◽  
Vol 17 (4) ◽  
pp. 977-1000 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Periodic Orbits ◽  
Invariant Measures ◽  
Hyperbolic Systems ◽  
Markov Property ◽  
Standard Technique ◽  
Equilibrium States ◽  
Convergence To Equilibrium ◽  
Invariant Density ◽  
Frobenius Operator ◽  
Uniformly Bounded

We study the convergence to equilibrium states for certain non-hyperbolic piecewise invertible systems. The multi-dimensional maps we shall consider do not satisfy Renyi's condition (uniformly bounded distortion for any iterates) and do not necessarily satisfy the Markov property. The failure of both conditions may cause singularities of densities of the invariant measures, even if they are finite, and causes a crucial difficulty in applying the standard technique of the Perron–Frobenius operator. Typical examples of maps we consider admit indifferent periodic orbits and arise in many contexts. For the convergence of iterates of the Perron–Frobenius operator, we study continuity of the invariant density.

scholarly journals Extremal dichotomy for uniformly hyperbolic systems

2015 ◽  
Vol 30 (4) ◽  
pp. 383-403 ◽  
Author(s):  
Maria Carvalho ◽  
Ana Cristina Moreira Freitas ◽  
Jorge Milhazes Freitas ◽  
Mark Holland ◽  
Matthew Nicol
Keyword(s):  
Hyperbolic Systems ◽  
Uniformly Hyperbolic Systems

Markov partitions and shadowing for non-uniformly hyperbolic systems with singularities

1992 ◽  
Vol 12 (3) ◽  
pp. 487-508 ◽  
Author(s):  
Tyll Krüger ◽  
Serge Troubetzkoy
Keyword(s):  
Large Class ◽  
Hyperbolic Systems ◽  
Markov Partitions ◽  
Uniformly Hyperbolic Systems ◽  
Dispersing Billiards

AbstractWe show the existence of countable Markov partitions for a large class of non-uniformly hyperbolic systems with singularities including dispersing billiards in any dimension.

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