A Morse equation in Conley's index theory for semiflows on metric spaces

1985 ◽  
Vol 5 (1) ◽  
pp. 123-143 ◽  
Author(s):  
Krzysztof P. Rybakowski ◽  
Eduard Zehnder
Keyword(s):  
Parabolic Equations ◽  
Metric Spaces ◽  
Index Theory ◽  
Invariant Set ◽  
Morse Decomposition ◽  
Cohomology Groups ◽  
Compact Spaces ◽  
Isolating Blocks ◽  
Isolated Invariant Set ◽  
Morse Decompositions

AbstractGiven a compact (two-sided) flow, an isolated invariant set S and a Morse-decomposition (M1, …, Mn) of S, there is a generalized Morse equation, proved by Conley and Zehnder, which relates the Alexander-Spanier cohomology groups of the Conley indices of the sets Mi and S with each other. Recently, Rybakowski developed the technique of isolating blocks and extended Conley's index theory to a class of one-sided semiflows on non-necessarily compact spaces, including e.g. semiflows generated by parabolic equations. Using these results, we discuss in this paper Morse decompositions and prove the above-mentioned Morse equation not only for arbitrary homology and cohomology groups, but also in this more general semiflow setting.

scholarly journals A refinement of the Conley index and an application to the stability of hyperbolic invariant sets

1987 ◽  
Vol 7 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Andreas Floer
Keyword(s):  
Continuous Flow ◽  
Index Theory ◽  
Conley Index ◽  
Invariant Set ◽  
Continuation Theorem ◽  
Invariant Sets ◽  
Additional Structure ◽  
Isolated Invariant Set ◽  
Continuation Property ◽  
The Stability

AbstractA compact and isolated invariant set of a continuous flow possesses a so called Conley index, which is the homotopy type of a pointed compact space. For this index a well known continuation property holds true. Our aim is to prove in this context a continuation theorem for the invariant set itself, using an additional structure. This refinement of Conley's index theory will then be used to prove a global and topological continuation-theorem for normally hyperbolic invariant sets.

scholarly journals FIXED POINT INDEX IN HYPERSPACES: A CONLEY-TYPE INDEX FOR DISCRETE SEMIDYNAMICAL SYSTEMS

2001 ◽  
Vol 64 (1) ◽  
pp. 191-204 ◽  
Author(s):  
F. R. RUIZ DEL PORTAL ◽  
J. M. SALAZAR
Keyword(s):  
Fixed Point ◽  
Fixed Point Index ◽  
Metric Spaces ◽  
Invariant Set ◽  
Invariant Sets ◽  
Point Index ◽  
Continuous Map ◽  
Isolated Invariant Set ◽  
Natural Way

Let X be a locally compact metric absolute neighbourhood retract for metric spaces, U ⊂ X be an open subset and f: U → X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the hyperspace of X. For any compact isolated invariant set, K ⊂ U, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors.

scholarly journals Morse decompositions and connection matrices

1988 ◽  
Vol 8 (8) ◽  
pp. 227-249 ◽  
Keyword(s):  
Metric Spaces ◽  
Morse Decomposition ◽  
Connection Matrix ◽  
Matrix Formalism ◽  
Compact Metric Spaces ◽  
Connection Matrices ◽  
Morse Decompositions ◽  
Morse Sets

AbstractThis paper surveys the work of Charles Conley and his students on Morse decompositions for flows on compact metric spaces, as well as the more recent development of the connection matrix formalism for detecting connections between the Morse sets of a Morse decomposition.

A Theorem on Nets

1966 ◽  
Vol 9 (3) ◽  
pp. 343-346
Author(s):  
M. Shimrat
Keyword(s):  
Metric Spaces ◽  
Countable Number ◽  
Simple Manner ◽  
Compact Spaces ◽  
Special Case

It is well-known that Tychonoff's theorem on the product of compact spaces may be proved, for the special case of a countable number of metric spaces X1, X2…, Xn,…, in the following simple manner.

SPHERICALIZATION AND FLATTENING PRESERVE UNIFORM DOMAINS IN NONLOCALLY COMPACT METRIC SPACES

2020 ◽  
pp. 1-22
Author(s):  
YAXIANG LI ◽  
SAMINATHAN PONNUSAMY ◽  
QINGSHAN ZHOU
Keyword(s):  
Metric Spaces ◽  
Complete Metric Spaces ◽  
Compact Spaces ◽  
Möbius Maps ◽  
Compact Metric Spaces ◽  
Uniform Domains ◽  
Invariant Properties

The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-Möbius maps preserve uniform domains in nonlocally compact spaces as well.

scholarly journals On Conley's fundamental theorem of dynamical systems

2004 ◽  
Vol 2004 (26) ◽  
pp. 1397-1401 ◽  
Author(s):  
M. R. Razvan
Keyword(s):  
Dynamical Systems ◽  
Fundamental Theorem ◽  
Index Theory ◽  
Conley Index ◽  
Morse Decomposition ◽  
Conley Index Theory

We generalize Conley's fundamental theorem of dynamical systems in Conley index theory. We also conclude the existence of a regular index filtration for every Morse decomposition.

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