Shift Equivalence and the Conley Index

Equivalence of Topological and Singular Transition Matrices in the Conley Index Theory

Detection of Renewal System Factors via the Conley Index

Let N be an isolating neighborhood for a map f . If we can decompose N into the disjoint union of compact sets N1 and N2, then we can relate the dynamics on the maximal invariant set InvN to the shift on two symbols by noting which component of N each iterate of a point x ∈ InvN lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to d...

Cocyclic Subshifts

Motivated by the computations in the theory of cohomological Conley index, cocyclic subshifts are the supports of locally constant matrix cocycles on the full shift over a nite alphabet. They properly generalize sooc systems and topological Markov chains; and, via the Wedderburn-Artin theory of nite-dimensional algebras, admit a similar structure theory with a spectral decomposition into mixing...

The Conley index for decompositions of isolated invariant sets

Let f be a continuous map of a locally compact metric space X into itself. Suppose that S is an isolated invariant set for f and a disjoint union of a fixed finite number of compact sets. We define an index of Conley type for isolated invariant sets admitting such a decomposition and prove some of its properties, which appear to be similar to that of the ordinary Conley index for maps. Our inde...

Symbolic Dynamics from Signed Matrices

We consider a method for assigning a sofic shift to a (not necessarily nonnegative integer) matrix by associating to it a directed graph with some vertices labelled 1 and the rest 2 (the decomposition of the vertices is arbitrary in applications the choice should be natural). We can detect positive topological entropy for this sofic shift by comparing the characteristic polynomial of the origin...