Pointwise Expansion Homeomorphisms

Reddy, William L.

doi:10.1112/jlms/s2-2.2.232

Cited by 29 publications

(14 citation statements)

References 3 publications

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“…We were not able to find a connection between isolated sets and point-wise expansivity [24] or h-expansivity [5].…”

mentioning

confidence: 59%

Isolated sets, catenary Lyapunov functions and expansive systems

Artigue¹

2016

TMNA

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Abstract. It is a paper about models for isolated sets and the construction of special hyperbolic Lyapunov functions. We prove that after a suitable surgery every isolated set is the intersection of an attractor and a repeller. We give linear models for attractors and repellers. With these tools we construct hyperbolic Lyapunov functions and metrics around an isolated set whose values along the orbits are catenary curves. Applications are given to expansive flows and homeomorphisms, obtaining, among other things, a hyperbolic metric on local cross sections for an arbitrary expansive flow on a compact metric space.

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“…We were not able to find a connection between isolated sets and point-wise expansivity [24] or h-expansivity [5].…”

mentioning

confidence: 59%

Isolated sets, catenary Lyapunov functions and expansive systems

Artigue¹

2016

TMNA

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“…Moreover, the set {(a, 1)Q p | a ∈ Q p } is uncountable as Q p is so. By Theorem 1 of [18], the D-action on H is not expansive. Therefore, the D-action on Sub Q 2 p is not expansive.…”

Section: Expansive Actions On Sub G Of Automorphisms Of Locally Compamentioning

confidence: 99%

“…Moreover, as T ((x n )) = ((x n−1 )) for all (x n ) ∈ T 1 , we get that T n (H) → {e} in Sub T 1 for all H ∈ H [26, Proposition 2.1]. By Theorem 1 of [18], T | T 1 is not expansive. This leads to a contradiction.…”

Section: Expansive Actions On Sub G Of Automorphisms Of Locally Compamentioning

confidence: 99%

Expansive actions of automorphisms of locally compact groups $${\varvec{G}}$$ on $$\hbox {Sub}_{{\varvec{G}}}$$

Prajapati

Shah

2020

Monatsh Math

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For a locally compact metrizable group G, we consider the action of Aut(G) on Sub G , the space of all closed subgroups of G endowed with the Chabauty topology. We study the structure of groups G admitting automorphisms T which act expansively on Sub G . We show that such a group G is necessarily totally disconnected, T is expansive and that the contraction groups of T and T −1 are closed and their product is open in G; moreover, if G is compact, then G is finite. We also obtain the structure of the contraction group of such T . For the class of groups G which are finite direct products of Q p for distinct primes p, we show that T ∈ Aut(G) acts expansively on Sub G if and only if T is expansive. However, any higher dimensional p-adic vector space Q n p , (n ≥ 2), does not admit any automorphism which acts expansively on Sub G .

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“…A bi-measurable map f on X is measure-expansive [14] if for any non-atomic Borel measure µ on X, we have µ(Γ δ (x)) = 0 for all x ∈ X. A bi-measurable map f on X is called pointwise expansive [16] if for each x ∈ X there is δ x > 0 (depending on x) such that Γ δx (x) = {x}.…”

Section: Preliminariesmentioning

confidence: 99%

Measure Expansivity and Specification for Pointwise Dynamics

Das

Khan

Das

2019

Bull Braz Math Soc, New Series

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We introduce pointwise measure expansivity for bi-measurable maps. We show through examples that this notion is weaker than measure expansivity. In spite of this fact, we show that many results for measure expansive systems hold true for pointwise systems as well. Then, we study the concept of mixing, specification and chaos at a point in the phase space of a continuous map. We show that mixing at a shadowable point is not sufficient for it to be a specification point, but mixing of the map force a shadowable point to be a specification point. We prove that periodic specification points are Devaney chaotic point. Finally, we show that existence of two distinct specification points is sufficient for a map to have positive Bowen entropy. (2010): 54H20, 37C50, 37B40 Mathematics Subject Classifications

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Pointwise Expansion Homeomorphisms

Cited by 29 publications

References 3 publications

Isolated sets, catenary Lyapunov functions and expansive systems

Isolated sets, catenary Lyapunov functions and expansive systems

Expansive actions of automorphisms of locally compact groups $${\varvec{G}}$$ on $$\hbox {Sub}_{{\varvec{G}}}$$

Measure Expansivity and Specification for Pointwise Dynamics

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