The Conley index and symbolic dynamics

The Conley Index and Symbolic Dynamics

Suppose that N is an isolating neighborhood for S with respect to a continuous map f which decomposes into a disjoint union of nitely many compact sets. We establish a relationship between the Conley indices of subsets of S which appear in a natural way when studying the behavior of f from the viewpoint of symbolic dynamics. The relationship is then used to prove a simple criterion for chaos in...

The Square of a Map, Symbolic Dynamics, and the Conley Index

We consider a map f from a locally compact metric space to itself, and use the discrete Conley index to study the difference between the local dynamics of f and f. In particular, we present a method, based on work by Mischaikow, Szymczak, et al., for detecting positive entropy symbolic dynamics by measuring the difference between Conley indices for f and f.

Reconstructing the Global Dynamics of Attractors via the Conley Index

Given an unknown attractor A in a continuous dy-namical system, how we can discover the topology and dynamics of A? As a practical matter, how can we do so from only a nite amount of information? One way of doing so is to produce a semi-conjugacy from A onto a model system M whose topology and dynamics are known. The complexity of M then provides a lower bound for the complexity of A. The Conle...

Shift Equivalence and the Conley Index

In this paper we introduce filtration pairs for isolated invariant sets of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Lastly, we define partially ordered Morse decom...

The Conley Index and Rigorous Numerics