Automorphisms of hyperbolic dynamical systems and $K_ 2$

Abstract:

Let $\sigma :\Sigma \to \Sigma$ be a subshift of finite type and $\operatorname {Aut} (\sigma )$ be the group of homeomorphisms of $\Sigma$ which commute with $\sigma$. In [Wl], Wagoner constructs an invariant for the group $\operatorname {Aut} (\sigma )$ using $K$-theoretic methods. Smooth hyperbolic dynamical systems can be modeled by subshifts of finite type over the nonwandering sets. In this paper we extend Wagoner’s construction to produce an invariant on the group of homeomorphisms of a smooth manifold which commute with a fixed hyperbolic diffeomorphism. We then proceed to show that this dynamical invariant can be calculated (at least $\bmod 2$) from the homology groups of the manifold and the action of the diffeomorphism and the homeomorphisms on the homology groups.