An obstruction to the existence of certain dynamics in surface diffeomorphisms

Abstract

AbstractLet M be a two-dimensional, compact manifold and g:Μ→ΜM be a diffeomorphism with a hyperbolic chain recurrent set. We find restrictions on the reduced zeta function p(t) of anyzero-dimensional basic set of g. If deg (p(t)) is odd, then p(1) = 0 (in ). Since there are infinitely many subshifts of finite type whose reduced zeta functions do not satisfy these restrictions, there are infinitely many subshifts which cannot be basic sets for any diffeomorphism of any surface.