Syndetically proximal pairs
type: Item
instance of: scholarly article
title: Syndetically proximal pairs
zbMATH Open document ID: 1225.54011
author: T. K. Subrahmonian Moothathu
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-04-20T00:00:00Z
review text: Consider a dynamical system formed by the complex \((X,f)\) where \(X\) is a compact metric space and \(f: X\to X^-\) a continuous map. The aim of this paper is to analyze the concept of syndetically proximal as an extension of the classical notion. Sufficient conditions for having syndetically pairs are given. It is proven that any interval map \(f\) with positive topological entropy has a syndetically scrambled Cantor set and an uncountable syndetically scrambled set invariant under some power of \(f\). Moreover, an improvement is obtained of the classical result: ``If \(f\) is an interval map with positive topological entropy and \(m\geq 2\) then there is \(n\in\mathbb{N}\) such that the one-sided full shift on \(m\) symbols is semi-conjugate to a subsystem of \(f^{2^{n}}\)'' by stating conjugacy instead of semi-conjugacy.
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 5880879
zbMATH Keywords: syndetically proximal relation
zbMATH Keywords: scrambled set
zbMATH Keywords: topological entropy
zbMATH Keywords: minimal point
zbMATH Keywords: transitivity
zbMATH Keywords: weak mixing
zbMATH Keywords: interval maps
zbMATH Keywords: subshifts
reviewed by: Juan L. G. Guirao
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.01.060
OpenAlex ID: W2091495376
cites work: Li–Yorke sensitivity
cites work: On Li-Yorke pairs
cites work: Dynamics in one dimension
cites work: Proximity Relations in Transformation Groups
cites work: On the invariance of Li–Yorke chaos of interval maps
cites work: Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation
cites work: Homeomorphisms with the whole compacta being scrambled sets
cites work: Devaney's chaos or 2-scattering implies Li-Yorke's chaos
cites work: Q3482542
cites work: A characterization of chaos
cites work: The return times set and mixing for measure preserving transformations
cites work: An Introduction to Symbolic Dynamics and Coding
cites work: Category version of the Poincaré recurrence theorem
cites work: Quantitative views of recurrence and proximality
cites work: Invariant scrambled sets and distributional chaos
cites work: Q3931654
DOI: 10.1016/J.JMAA.2011.01.060
description: scientific article; zbMATH DE number 5880879