scholarly journals A Markov partition that reflects the geometry of a hyperbolic toral automorphism

2002 ◽  
Vol 354 (7) ◽  
pp. 2849-2863 ◽  
Author(s):  
Anthony Manning
Keyword(s):  
Markov Partition ◽  
Toral Automorphism ◽  
Hyperbolic Toral Automorphism

Clustering of periodic orbits of a hyperbolic automorphism on the torus T2

2021 ◽  
Vol 15 (1) ◽  
pp. 51-60
Author(s):  
Minh Hien Huynh ◽  
◽  
Van Nam Vo ◽  
Tinh Le ◽  
Thi Dai Trang Nguyen
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Toral Automorphism ◽  
Number Of Clusters ◽  
Periodic Sequences ◽  
Axiom A ◽  
Hyperbolic Automorphism ◽  
Hyperbolic Toral Automorphism ◽  
Symbolic Dynamical System

This paper deals with clustering of periodic orbits of the hyperbolic toral automorphism induced by matrix A. We prove that Ta satisfies the Axiom A. The clustering of periodic orbits of Ta is ivestigated via the notion of 'p-closeness' of periodic sequences of the respective symbolic dynamical system. We also provide the number of clusters of periodic sequences with given periods in the case of 2-closeness.

scholarly journals The orbit of a Hölder continuous path under a hyperbolic toral automorphism

1983 ◽  
Vol 3 (3) ◽  
pp. 345-349 ◽  
Author(s):  
M. C. Irwin
Keyword(s):  
Orbit Closure ◽  
Continuous Path ◽  
Toral Automorphism ◽  
Hölder Continuous ◽  
Linear Automorphism ◽  
Hyperbolic Toral Automorphism

AbstractLet f:T3→T3 be a hyperbolic toral automorphism lifting to a linear automorphism with real eigenvalues. We prove that there is a Hölder continuous path in T3 whose orbit-closure is 1-dimensional. This strengthens results of Hancock and Przytycki concerning continuous paths, and contrasts with results of Franks and Mañé concerning rectifiable paths.

scholarly journals Hölder continuous paths and hyperbolic toral automorphisms

1986 ◽  
Vol 6 (2) ◽  
pp. 241-257 ◽  
Author(s):  
M. C. Irwin
Keyword(s):  
Orbit Closure ◽  
Continuous Path ◽  
Toral Automorphism ◽  
Hölder Continuous ◽  
Hyperbolic Toral Automorphism ◽  
Toral Automorphisms

AbstractLet f:Tn→Tn (n ≥ 3) be a hyperbolic toral automorphism. Let A be the set of α > 0 such that there is a Hölder continuous path of index α in Tn with 1-dimensional orbit-closure under f We prove that α0 = sup A can be expressed in terms of the eigenvalues of f and that α0 ∈ A if and only if α0 < 1.

scholarly journals Geometric realization for substitution tilings

2012 ◽  
Vol 34 (2) ◽  
pp. 457-482 ◽  
Author(s):  
MARCY BARGE ◽  
JEAN-MARC GAMBAUDO
Keyword(s):  
Geometric Realization ◽  
Toral Automorphism ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Substitution Tilings ◽  
Hyperbolic Toral Automorphism ◽  
Tiling Space ◽  
The One ◽  
Cech Cohomology ◽  
Family Condition

AbstractGiven an n-dimensional substitution Φ whose associated linear expansion Λ is unimodular and hyperbolic, we use elements of the one-dimensional integer Čech cohomology of the tiling space ΩΦ to construct a finite-to-one semi-conjugacy G:ΩΦ→𝕋D, called a geometric realization, between the substitution induced dynamics and an invariant set of a hyperbolic toral automorphism. If Λ satisfies a Pisot family condition and the rank of the module of generalized return vectors equals the generalized degree of Λ, G is surjective and coincides with the map onto the maximal equicontinuous factor of the ℝn-action on ΩΦ. We are led to formulate a higher-dimensional generalization of the Pisot substitution conjecture: if Λ satisfies the Pisot family condition and the rank of the one-dimensional cohomology of ΩΦ equals the generalized degree of Λ, then the ℝn-action on ΩΦhas pure discrete spectrum.

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.

The construction of a quantum Markov partition

1999 ◽  
Vol 32 (42) ◽  
pp. 7273-7286 ◽  
Author(s):  
Raúl O Vallejos ◽  
Marcos Saraceno
Keyword(s):  
Markov Partition

Inductive Limit Toral Automorphisms of Irrational Rotation Algebras

2001 ◽  
Vol 44 (3) ◽  
pp. 335-336
Author(s):  
P. J. Stacey
Keyword(s):  
Inductive Limit ◽  
Continuous Functions ◽  
Irrational Rotation ◽  
Space Of Continuous Functions ◽  
Toral Automorphism ◽  
Matrix Algebras ◽  
Toral Automorphisms

AbstractIrrational rotation C*-algebras have an inductive limit decomposition in terms of matrix algebras over the space of continuous functions on the circle and this decomposition can be chosen to be invariant under the flip automorphism. It is shown that the flip is essentially the only toral automorphism with this property.

Stationary policies with Markov partition property

2010 ◽  
Vol 13 (6) ◽  
pp. 1323-1341 ◽  
Author(s):  
John E. Goulionis ◽  
Dimitrios I. Stengos ◽  
George Tzavelas
Keyword(s):  
Markov Partition ◽  
Partition Property
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