Title: Non-Smooth Dynamical Systems that Exhibit Hyperbolic Behavior
Speaker: Professor Marian Gidea
Speaker Info: Loyola University of Chicago
The Conley index theory can be viewed as an extension of Morse theory to arbitrary flows defined on locally compact metric spaces. The objects of primary interest in Conley's approach are invariant sets and their isolating neighborhoods.Date: Tuesday, February 24, 1998
There are some similarities between invariant sets and hyperbolic sets but also some major differences. On the one hand, the Conley index of an invariant set is stable under small perturbations; on the other hand, the structure of the set and the nearby dynamics can be very complicated.
In order to get a better understanding of the (possible) chaotic dynamics on invariant sets, we construct a Conley index for non-invariant sets (subsets of a larger invariant set). We use this index to study the asymptotic behavior of some orbits relative to the reference orbit of the non-invariant set and to detect symbolic dynamics.
We can extend the concept of hyperbolic system to non-smooth dynamical systems in terms of this new Conley index. Stability of the system, shadowing lemma and existence of partitions of Markov type can be proved under some topological assumptions. Our examples include topological horseshoes (with or without singularities), non-uniformly hyperbolic dynamical systems with singularities, Julia sets.