scholarly journals Quenched CLT for random toral automorphism

2009 ◽  
Vol 24 (2) ◽  
pp. 331-348 ◽  
Author(s):  
Arvind Ayyer ◽  
◽  
Carlangelo Liverani ◽  
Mikko Stenlund ◽  
◽  
...  
Keyword(s):  
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.

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.

scholarly journals Some compact invariant sets for hyperbolic linear automorphisms of torii

1988 ◽  
Vol 8 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Albert Fathi
Keyword(s):  
Hausdorff Dimension ◽  
Homology Group ◽  
Invariant Set ◽  
Finite Union ◽  
Invariant Sets ◽  
Pisot Number ◽  
Toral Automorphism ◽  
Lower Capacity

AbstractIf the action induced by a pseudo-Anosov map on the first homology group is hyperbolic, it is possible, by a theorem of Franks, to find a compact invariant set for the toral automorphism associated with this action. If the stable and unstable foliations of the Pseudo-Anosov map are orientable, we show that the invariant set is a finite union of topological 2-discs. Using some ideas of Urbański, it is possible to prove that the lower capacity of the associated compact invariant set is >2; in particular, the invariant set is fractal. When the dilatation coefficient is a Pisot number, we can compute the Hausdorff dimension of the compact invariant set.

scholarly journals Sets of all periodic points of a toral automorphism

2010 ◽  
Vol 366 (1) ◽  
pp. 367-371 ◽  
Author(s):  
I. Subramania Pillai ◽  
K. Ali Akbar ◽  
V. Kannan ◽  
B. Sankararao
Keyword(s):  
Periodic Points ◽  
Toral Automorphism

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.

The topological entropy of non-dense orbits and generalized Schmidt games

2017 ◽  
Vol 39 (2) ◽  
pp. 500-530
Author(s):  
WEISHENG WU
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Metric Entropy ◽  
Toral Automorphism ◽  
Partially Hyperbolic ◽  
Dense Orbits ◽  
Bounded Below ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism

We generalize the notion of Schmidt games to the setting of the general Caratheódory construction. The winning sets for such generalized Schmidt games usually have large corresponding Caratheódory dimensions (e.g., Hausdorff dimension and topological entropy). As an application, we show that for every $C^{1+\unicode[STIX]{x1D703}}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$ satisfying certain technical conditions, the topological entropy of the set of points with non-dense forward orbits is bounded below by the unstable metric entropy (in the sense of Ledrappier–Young) of certain invariant measures. This also gives a unified proof of the fact that the topological entropy of such a set is equal to the topological entropy of $f$, when $f$ is a toral automorphism or the time-one map of a certain non-quasiunipotent homogeneous flow.

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