Rigorous Symbolic Dynamics and Entropy via Conley Index Theory
Rafael Frongillo
Department of Computer Science, University of Colorado Boulder
Date and time:
Thursday, November 19, 2015 - 2:00pm
Location:
ECCR 257
Abstract:
Computers have helped us prove many theorems, which like the four-color theorem are mostly combinatorial or discrete in nature. For domains like dynamical systems, however, computer-assisted proofs are more challenging; by definition, we must convert the continuous problem into a discrete one, in such a way as to (provably) preserve enough structure of the original system that we can still make rigorous statements. Conley index theory, a generalization of Morse theory using algebraic topology, is a tool that allows one to do just that, and has been used in a computational framework to prove the existence of dynamics of various types.
When searching for highly complicated dynamics, however, the Conley index may also become highly complicated, and difficult to interpret. I will present an automated approach to processing Conley index information for discrete-time dynamical systems, joint work with Sarah Day and Rodrigo Treviño. This approach produces a topologically semi-conjugate symbolic system whose entropy serves as a lower bound for the entropy of the system under study. I will describe several applications and modifications of this approach, culminating in recent work with Sarah Day which produces symbolic systems that in some sense capture as much information from the index as possible, in some cases leading to substantial improvements in entropy lower bounds.