A ‘horseshoe’ theorem in symbolic dynamics via single sequence techniques.
Sep 26 2013 – 3:00pm – 4:00 pm
Daniel J Thompson (The Ohio State University)
For a broad class of symbolic dynamical systems without the Markov property, including the coding spaces of many piecewise continuous interval maps, we show how to approximate an arbitrary ergodic measure with a measure of almost the same entropy supported on a sofic subshift. This is interpreted as a symbolic analogue of a `hyperbolic horseshoe’ theorem. In addition to the intrinsic interest of this result as a structure theorem, it can be a useful tool in large deviations theory and multifractal analysis. I will discuss two ways to establish this result, both based on surgery on a single generic orbit. One proof is based on Ornstein’s d bar metric, and the other is based on the theory of Kolmogorov complexity. Both techniques can be explained in a simple and intuitive way.