scholarly journals Analogues of Auslander–Yorke theorems for multi-sensitivity

2016 ◽  
Vol 38 (2) ◽  
pp. 651-665 ◽  
Author(s):  
WEN HUANG ◽  
SERGIĬ KOLYADA ◽  
GUOHUA ZHANG
Keyword(s):  
Metric Spaces ◽  
Minimal System ◽  
Compact Metric Spaces ◽  
Dichotomy Theorem ◽  
One To One ◽  
Transitive System

We study multi-sensitivity and thick sensitivity for continuous surjective selfmaps on compact metric spaces. Our main result states that a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor. This is an analog of the Auslander–Yorke dichotomy theorem: a minimal system is either sensitive or equicontinuous. Furthermore, we introduce the concept of a syndetically equicontinuous point, and we prove that a transitive system is either thickly sensitive or contains syndetically equicontinuous points, which is a refinement of another well-known result of Akin, Auslander and Berg.

The structural stability of topological cocycles

1999 ◽  
Vol 19 (5) ◽  
pp. 1309-1324
Author(s):  
K. M. MADDEN ◽  
N. G. MARKLEY ◽  
M. SEARS
Keyword(s):  
Structural Stability ◽  
Metric Spaces ◽  
Time Change ◽  
Dense Orbit ◽  
Small Perturbations ◽  
Suspension Flows ◽  
Compact Metric Spaces ◽  
Higher Dimensional ◽  
One To One

Cocycles of $Z^m$ actions on compact metric spaces can be used to construct $R^m$ actions or flows, called suspension flows. A suspension provides a higher-dimensional analog to the familiar flow under a function and we look to this construction as a way of generating interesting $R^m$ flows. Even more importantly, an $R^m$ flow with a free dense orbit has an almost one-to-one extension which is a suspension [6] and thus suspensions can be used to model general $R^m$ flows. In this paper we examine the sensitivity of the suspension construction to small perturbations in the cocycle. Theorem 4.7 establishes the fact that two cocycles that are sufficiently close yield suspensions that are isomorphic up to a time change.

scholarly journals The Blum-Hanson Property

2019 ◽  
Vol 6 (1) ◽  
pp. 92-105
Author(s):  
Sophie Grivaux
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Separable Banach Space ◽  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Theoretic Motivation

AbstractGiven a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means{1 \over N}\sum\limits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.

DIAGONALIZATION MODULO NORM IDEALS AND HAUSDORFF DIMENSIONALITY

2000 ◽  
Vol 11 (08) ◽  
pp. 1057-1078
Author(s):  
JINGBO XIA
Keyword(s):  
Hausdorff Measure ◽  
Metric Spaces ◽  
Adjoint Operator ◽  
Trace Class ◽  
Multiplication Operators ◽  
Von Neumann ◽  
Dimensional Hausdorff Measure ◽  
One Dimensional ◽  
Compact Metric Spaces ◽  
The One

Kuroda's version of the Weyl-von Neumann theorem asserts that, given any norm ideal [Formula: see text] not contained in the trace class [Formula: see text], every self-adjoint operator A admits the decomposition A=D+K, where D is a self-adjoint diagonal operator and [Formula: see text]. We extend this theorem to the setting of multiplication operators on compact metric spaces (X, d). We show that if μ is a regular Borel measure on X which has a σ-finite one-dimensional Hausdorff measure, then the family {Mf:f∈ Lip (X)} of multiplication operators on T2(X, μ) can be simultaneously diagonalized modulo any [Formula: see text]. Because the condition [Formula: see text] in general cannot be dropped (Kato-Rosenblum theorem), this establishes a special relation between [Formula: see text] and the one-dimensional Hausdorff measure. The main result of the paper is that such a relation breaks down in Hausdorff dimensions p>1.

scholarly journals Classification Of Locally 2-Connected Compact Metric Spaces

COMBINATORICA ◽  
2004 ◽  
Vol 25 (1) ◽  
pp. 85-103 ◽  
Author(s):  
Carsten Thomassen
Keyword(s):  
Metric Spaces ◽  
Compact Metric Spaces

scholarly journals Stationary points for set-valued mappings on two metric spaces

2001 ◽  
Vol 28 (7) ◽  
pp. 427-432
Author(s):  
Zeqing Liu ◽  
Qingtao Liu ◽  
Shin Min Kang
Keyword(s):  
Stationary Point ◽  
Metric Spaces ◽  
Stationary Points ◽  
Compact Metric Spaces ◽  
Set Valued Mappings

We give stationary point theorems of set-valued mappings in complete and compact metric spaces. The results in this note generalize a few results due to Fisher.

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