scholarly journals Remarks on the region of attraction of an isolated invariant set

2006 ◽  
Vol 104 (2) ◽  
pp. 157-167 ◽  
Author(s):  
Konstantin Athanassopoulos
Keyword(s):  
Invariant Set ◽  
Region Of Attraction ◽  
Isolated Invariant Set

scholarly journals Dynamical systems, shape theory and the Conley index

1988 ◽  
Vol 8 (8) ◽  
pp. 375-393 ◽  
Keyword(s):  
Dynamical Systems ◽  
Shape Index ◽  
Conley Index ◽  
Invariant Set ◽  
Discrete Time Systems ◽  
Isolated Invariant Set ◽  
The One ◽  
One Point Compactification ◽  
Time Systems ◽  
Construction Works

AbstractThe Conley index of an isolated invariant set is defined only for flows; we construct an analogue called the ‘shape index’ for discrete dynamical systems. It is the shape of the one-point compactification of the unstable manifold of the isolated invariant set in a certain topology which we call its ‘intrinsic’ topology (to distinguish it from the ‘extrinsic’ topology which it inherits from the ambient space). Like the Conley index, it is invariant under continuation. A key point is the construction of a certain ‘index category’ associated with the isolated invariant set; this construction works equally well for flows or discrete time systems, and its properties imply the basic properties of both the Conley index and the shape index.

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.

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 Open index pairs, the fixed point index and rationality of zeta functions

1990 ◽  
Vol 10 (3) ◽  
pp. 555-564 ◽  
Author(s):  
Marian Mrozek
Keyword(s):  
Fixed Point ◽  
Fixed Point Index ◽  
Zeta Functions ◽  
Invariant Set ◽  
Lefschetz Number ◽  
Point Index ◽  
Index Pair ◽  
Index Pairs ◽  
Isolated Invariant Set

AbstractWe define open index pairs of an isolated invariant set, prove their existence and compute the fixed point index of an isolating neighbourhood in terms of the Lefschetz number of a certain map associated with the open index pair. We use this to establish rationality of zeta functions and Lefschetz zeta functions.

Some properties of the isolated invariant set of a flow

2003 ◽  
Vol 16 (5) ◽  
pp. 685-689 ◽  
Author(s):  
Hu Fan-Nu ◽  
Zheng Zuo-Huan
Keyword(s):  
Invariant Set ◽  
Isolated Invariant Set

Equivariant Morse equation

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Marcin Styborski
Keyword(s):  
Lie Group ◽  
Conley Index ◽  
Invariant Set ◽  
Compact Lie Group ◽  
Multiplicity Results ◽  
Equivariant Degree ◽  
Isolated Invariant Set

AbstractThe paper is concerned with the Morse equation for flows in a representation of a compact Lie group. As a consequence of this equation we give a relationship between the equivariant Conley index of an isolated invariant set of the flow given by .x = −∇f(x) and the gradient equivariant degree of ∇f. Some multiplicity results are also presented.

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