Markov Partitions: from Decimal Expansions to Nilmanifolds
Talk by Dr. Jarek Kwapisz (Mathematical Sciences, MSU)
02/01/2021 4:10-5:15pm Zoom Talk
A deterministic dynamical system (like the weather) can be chaotic: its long term behavior is so unpredictable (under tiniest of the initial data uncertainty) that it is best understood as a stochastic process. The proverbial rolling of a dice (repeatedly) is one such process; and replacing the dice with a finite state automaton yields ubiquitous Markov chains (or sofic shifts). Finding the right automaton for a given dynamical system can be tricky and involves partitioning the dynamical space into carefully designed (fractal) subsets called Markov boxes. For the flagship class of (uniformly hyperbolic) chaotic systems called Anosov maps, existence of such partitions has been known for over 40 years but their design methods lagged and only touched the simplest subclass, the maps of tori. We develop a construction applicable to all known Anosov maps (up to a covering). Its validation is the first ever Markov partition for Smale's famous 1967 example on a six dimensional nilmanifold, the simplest non-toral example. I will explain the key ideas and how the whole story is a far reaching extension of the concept of the ordinary decimal expansion.