Morse Inequalities and Zeta Functions

Morse inequalities for diffeomorphisms of a compact manifold were first proved by Smale [21] under the assumption that the nonwandering set is finite. We call these the integral Morse inequalities. They were generalized by Zeeman in an unpublished work cited in [25] to diffeomorphisms with a hyperbolic chain recurrent set that is axiom-A-diffeomorphisms which satisfy the no cycle condition. For the same class of diffeomorphisms Franks derived in [7] and [8] the polynomial Morse inequalities. These contain the integral Morse inequalities as a special case. In this paper we generalize the polynomial Morse inequalities to Morse decompositions of isolated invariant sets. To explain the relationship between the integral and the polynomial Morse inequalities we consider a diffeomorphism f : M →M of a compact manifold M with a hyperbolic chain recurrent set Rf . By the shadowing lemma Rf decomposes into finitely many chain transitive components {Λp}p∈P satisfying the no cycle condition. Under these assumptions all the periodic points of f are nondegenerate. Counting the periodic points in the basic set Λp algebraically gives the homology zeta function [22]