scholarly journals Refinements of topological invariants of flows

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tomoo Yokoyama
Keyword(s):  
Metric Spaces ◽  
Orbit Space ◽  
Singular Points ◽  
Connected Components ◽  
Topological Spaces ◽  
Topological Invariants ◽  
Orbit Spaces ◽  
Compact Metric Spaces ◽  
Hamiltonian Flows ◽  
Closed Orbits

<p style='text-indent:20px;'>We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces. In particular, the abstract weak orbit spaces of flows on topological spaces are refinements of Morse graphs of flows on compact metric spaces, Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces, and the CW decompositions which consist of the unstable manifolds of singular points for Morse flows on closed manifolds. Though the CW decomposition of a Morse flow is finite, the intersection of the unstable manifold and the stable manifold of closed orbits need not consist of finitely many connected components. Therefore we study the finiteness. Moreover, we consider when the time-one map reconstructs the topology of the original flow. We show that the orbit space of a Hamiltonian flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map by taking an arbitrarily small reparametrization and that the abstract weak orbit spaces of a Morse flow on a compact manifold and the time-one map are homeomorphic. In addition, we state examples whose Morse graphs are singletons but whose abstract weak orbit spaces are not singletons.</p>

Proximate topology and shape theory

1995 ◽  
Vol 125 (3) ◽  
pp. 595-615 ◽  
Author(s):  
Zvonko Čerin
Keyword(s):  
Metric Spaces ◽  
Topological Spaces ◽  
Shape Theory ◽  
Homotopy Classes ◽  
New Approach ◽  
Compact Metric Spaces ◽  
Shape Category

Most of the development of shape theory was in the so-called outer shape theory, where the shape of spaces is described with the help of some outside objects.This paper belongs to the so-called inner shape theory, in which the shape of spaces is described intrinsically without the use of any outside gadgets. We give a description of shape theory that does not need absolute neighbourhood retracts. We prove that the category ℋN whose objects are topological spaces and whose morphisms are proximate homotopy classes of proximate nets is naturally equivalent to the shape category h. The description of the category ℋN for compact metric spaces was given earlier by José M. R. Sanjurjo. We also give three applications of this new approach to shape theory.

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 A contribution to the study of fuzzy metric spaces

2001 ◽  
Vol 2 (1) ◽  
pp. 63 ◽  
Author(s):  
Almanzor Sapena Piera
Keyword(s):  
Metric Spaces ◽  
Fuzzy Theory ◽  
Topological Spaces ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric

We give some examples nad properties of fuzzy metric spaces, in the sense of George and Veramani, and characterize the To topological spaces which admit a compatible uniformity that has a countable transitive base, in terms of the fuzzy theory.

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