Homoclinic points of algebraic $\mathbb {Z}^d$-actions

Lind, Douglas; Schmidt, Klaus

doi:10.1090/s0894-0347-99-00306-9

Cited by 114 publications

(91 citation statements)

References 18 publications

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“…Then the automorphism on the torus T d defined by T x = Bx mod Z d has no homoclinic point (see Example 3.4. in [19]). Consequently, for any integer m ≥ 1, the power T m has no homoclinic point.…”

Section: Lemma 42 ([5]mentioning

confidence: 98%

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Bohr chaoticity of topological dynamical systems

Fan¹,

Fan²,

Ryzhikov³

et al. 2021

Preprint

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We introduce the notion of Bohr chaoticity, which is a topological invariant for topological dynamical systems, and which is opposite to the property required by Sarnak's conjecture. We prove the Bohr chaoticity for all systems which have a horseshoe and for all toral affine dynamical systems of positive entropy, some of which don't have a horseshoe. But uniquely ergodic dynamical systems are not Bohr chaotic.We say that (f, x) verifying (1.3) is a pair correlated to (w n ) for the system (X, T ).

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Section: Lemma 42 ([5]mentioning

confidence: 98%

“…According to Lind and Schmidt (cf. Example 3.4 in [19]), any ergodic irreducible partially hyperbolic toral automorphism T A defined by a matrix A ∈ GL(d, Z) has no non-trivial homoclinic point. Here is an example:…”

Section: Below)mentioning

confidence: 99%

Bohr chaoticity of topological dynamical systems

Fan¹,

Fan²,

Ryzhikov³

et al. 2021

Preprint

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“…Remark that E s ∩ Z n = {0}, E u ∩ Z n = {0}, and both subgroups π(E s ) and π(E u ), as well as the homoclinic group ∆ α = π(E s ) ∩ π(E u ) induced by α, are dense in R n /Z n . Moreover, the system Z α R n /Z n generated by α is topologically free ( [11]).…”

Section: Expansive Automorphism Actions On Compact Groupsmentioning

confidence: 99%

Continuous orbit equivalence up to equivalence relations

Qiang¹,

Hou²

2021

Preprint

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We introduce notions of continuous orbit equivalence and strong (respective, weak) continuous orbit equivalence for automorphism systems of étale equivalence relations, and characterize them in terms of the semi-direct product groupoids, as well as their reduced groupoid C * -algebras with canonical Cartan subalgebras. In particular, we study topological rigidity of expansive automorphism actions on compact (connected) metrizable groups.

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“…There is also a cumulative development on the expansive actions by continuous automorphisms on compact metrizable abelian groups appealing tools from commutative algebras, operator algebras, etc. [10,5,12,3].…”

Section: Introductionmentioning

confidence: 99%

The non-coexistence of distality and expansivity for group actions on infinite compacta

Liang¹,

Shi²,

Xie³

et al. 2021

Preprint

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Let X be a compact metric space and G a finitely generated group. Suppose φ : G → Homeo(X) is a continuous action. We show that if φ is both distal and expansive, then X must be finite. A counterexample is constructed to show the necessity of finite generation condition on G. This is also a supplement to a result due to Auslander-Glasner-Weiss which says that every distal action by a finitely generated group on a zerodimensional compactum is equicontinuous.

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Homoclinic points of algebraic $\mathbb {Z}^d$-actions

Cited by 114 publications

References 18 publications

Bohr chaoticity of topological dynamical systems

Bohr chaoticity of topological dynamical systems

Continuous orbit equivalence up to equivalence relations

The non-coexistence of distality and expansivity for group actions on infinite compacta

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