toral automorphism
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2021 ◽  
Vol 15 (1) ◽  
pp. 51-60
Author(s):  
Minh Hien Huynh ◽  
◽  
Van Nam Vo ◽  
Tinh Le ◽  
Thi Dai Trang Nguyen

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.



2017 ◽  
Vol 39 (2) ◽  
pp. 500-530
Author(s):  
WEISHENG WU

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.





2013 ◽  
Vol 75 (1-2) ◽  
pp. 1-6 ◽  
Author(s):  
Hao-Kuan Tso


2013 ◽  
Vol 33 (3) ◽  
pp. 896-933 ◽  
Author(s):  
ANDY HAMMERLINDL

AbstractIf a partially hyperbolic diffeomorphism on a torus of dimension $d\geq 3$ has stable and unstable foliations which are quasi-isometric on the universal cover, and its centre direction is one-dimensional, then the diffeomorphism is leaf conjugate to a linear toral automorphism. In other words, the hyperbolic structure of the diffeomorphism is exactly that of a linear, and thus simple to understand, example. In particular, every partially hyperbolic diffeomorphism on the 3-torus is leaf conjugate to a linear toral automorphism.



2012 ◽  
Vol 34 (2) ◽  
pp. 457-482 ◽  
Author(s):  
MARCY BARGE ◽  
JEAN-MARC GAMBAUDO

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.



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


2009 ◽  
Vol 24 (2) ◽  
pp. 331-348 ◽  
Author(s):  
Arvind Ayyer ◽  
◽  
Carlangelo Liverani ◽  
Mikko Stenlund ◽  
◽  
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2006 ◽  
Vol 26 (04) ◽  
pp. 961 ◽  
Author(s):  
MARCY BARGE ◽  
JAROSLAW KWAPISZ


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