Thursday, March 12, 2009 - 11:05am
1.5 hours (actually 80 minutes)
Motivated by Smale's work on smooth dynamical systems, David Ruelle introduced the notion of Smale spaces. These are topological dynamical systems which are hyperbolic in the sense of having local coordinates of contracting and expending directions. These include hyperbolic automorphisms of tori, but typically, the spaces involved have a fractal nature. An important subclass are the shifts of finite type which are symbolic systems described by combinatorial data. These are also precisely the Smale spaces which are totally disconnected. Rufus Bowen showed that every Smale space is the image of shift of finite type under a finite-to-one factor map. In the 1970's, Wolfgang Krieger introduced a beautiful invariant for shifts of finite type. The aim of this talk is to show how a refined version of Bowen's theorem may be used to extend Krieger's invariant to all (irreducible) Smale spaces. The talk will assume no prior knowledge of these topics - we begin with a discussion of Smale spaces and shifts of finite type and then move on to Krieger's invariant and its extension.