Conley decomposition for closed relations

Abstract

Abstract. This paper presents a theory of dynamics of closed relations on compact Hausdorff spaces. It contains an investigation of set valued maps and establishes generalizations for some topological aspects of dynamical systems theory, including recurrence, attractor-repeller structure and the Conley Decomposition Theorem.

Lattice Structures for Attractors II

“…The latter interpretation is useful from the perspective of algorithms, but treating F as a map provides intuition as to how to define important dynamical analogues in the discrete setting. This is discussed in detail in Section 2, but we also point out the work in [12,11] on closed relations. Given our focus on attractors there are three structures arising from combinatorial dynamics that are of particular interest: forward invariant sets, Invset + (X , F ) := {S ⊂ X | F (S) ⊂ (S)}; attracting sets, ASet(X , F ) := {U ⊂ X | ω(U, F ) ⊂ U}; and attractors, Att(X , F ) := {A ⊂ X | F (A) = A}.…”

Topological bifurcations of minimal invariant sets for set-valued dynamical systems

“…Under the additional assumption that M is an attractor in Proposition 4.1, i.e. A(M ) is a neighbourhood of M , the pair (M, M * ) is an attractor-repeller pair as discussed in [MW06]. This pair can be extended to obtain Morse decompositions, see [Li07].…”

“…The purpose of this section is to provide generalisations of attractor-repeller decompositions which were introduced in [MW06,Li07] for the study of Morse decompositions of set-valued dynamical systems. These generalisations are necessary for our purpose, because we deal with invariant sets rather than attractors, and they will be applied in Section 6 in the context of bifurcation theory.…”

Developments in fractal geometry

“…The following three definitions look almost identical to Definitions 2.2, 2.3, and 2.4, the crucial differences being that a Conley attractor A may be empty and orbits of sets under F : 2 X → 2 X are "absorbed" by A rather than converging, in a cascade, to A. A simple proof of Theorem 12.4 can be found in [36], but greater generality and other characterizations of Conley attractors and related entities appear in [81,91,92].…”

“…In this section we define the Conley attractor of an IFS, discuss the existence of attractorrepeller pairs (Theorem 12.5) and their relation to the dynamics of the IFS (Theorem 12.7), and explain Conley's "landscape picture" as it applies to an IFS (Theorem 12.9). Our presentation is a simplified version of the elegant work of McGehee and Wiandt [92,126] about iterated closed relations on compact Hausdorff spaces.…”