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.

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.

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 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

scholarly journals An analogue of Bauer’s theorem for closed orbits of skew products

2008 ◽  
Vol 28 (2) ◽  
pp. 535-546 ◽  
Author(s):  
WILLIAM PARRY ◽  
MARK POLLICOTT
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Hyperbolic Systems ◽  
Algebraic Number Theory ◽  
Skew Products ◽  
Closed Orbits

AbstractIn this article we prove an analogue of Bauer’s theorem from algebraic number theory in the context of 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 Generating special Markov partitions for hyperbolic toral automorphisms using fractals

1986 ◽  
Vol 6 (3) ◽  
pp. 325-333 ◽  
Author(s):  
Tim Bedford
Keyword(s):  
Transition Matrix ◽  
Markov Partition ◽  
Natural Conditions ◽  
Markov Partitions ◽  
Toral Automorphisms

AbstractWe show that given some natural conditions on a 3 × 3 hyperbolic matrix of integers A(det A = 1) there exists a Markov partition for the induced map A(x + ℤ3) = A(x)+ℤ3 on T3 whose transition matrix is (A−1)t. For expanding endomorphisms of T2 we construct a Markov partition so that there is a semiconjugacy from a full (one-sided) shift.

Symbolic Dynamics for IFS Attractors

Fractals ◽  
1997 ◽  
Vol 05 (02) ◽  
pp. 237-246 ◽  
Author(s):  
Sonya Bahar
Keyword(s):  
Symbolic Dynamics ◽  
Iterated Function System ◽  
Chaotic Regime ◽  
Function System ◽  
Control Parameters ◽  
Closed Orbits ◽  
Iterated Function ◽  
Parameter Values

It has recently been shown that a modified iterated function system (IFS) is capable of generating closed orbits which undergo bifurcation and transition to a chaotic regime as control parameters are varied.1,2 Here we show that driving such an IFS by a partition of itself creates maps which can be characterized by a symbolic dynamics. Forbidden words are determined for this dynamics under various parameter values, and the implications of this mapping are discussed.

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