Jim Wiseman

POSITIVELY EXPANSIVE HOMEOMORPHISMS OF COMPACT SPACES

We give a new and elementary proof showing that a homeomorphism f:X→X of a compact metric space is positively expansive if and only if X is finite.

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 N 1 and N 2 , then we can relate the dynamics on the maximal invariant set Inv N to the shift on two symbols by noting which component of N each iterate of a point x ∈ Inv N lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to N. In essence, we measure the difference between the Conley index of Inv N and the sum of the indices of Inv N 1 and Inv N 2 .

SYMBOLIC DYNAMICS FOR NONHYPERBOLIC SYSTEMS

We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possesses an index system. Because of their topological robustness, they can be used to obtain rigorous results from computer approximations of a dynamical system.

The generalized recurrent set and strong chain recurrence

Fathi and Pageault have recently shown a connection between Auslanders generalized recurrent set

How Should Self-Driving Cars Drive?

GR(f)

Noting the Difference: Musical Scales and Permutations

and Eastons strong chain recurrent set. We study

Chain recurrence rates and topological entropy

GR(f)

A fixed point theorem for bounded dynamical systems

by examining that connection in more detail, as well as connections with other notions of recurrence. We give equivalent definitions that do not refer to a metric. In particular, we show that

Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces

GR(f^k)=GR(f)

Addendum to: ``A fixed point theorem for bounded dynamical systems'' [Illinois J. Math. 46 (2002), no. 2, 491--495]

for any