Morse inequalities for a dynamical system

Morse Inequalities for a Dynamical System

Witten’s Proof of Morse Inequalities

Both properties do not depend on the choice of coordinates. The index ind (x) is the number of negative eigenvalues of Hess (f) (x). Let mp = mp (f) be the number of critical points of index p. Let bp = bp (M) = dimH (M) be the dimension of the p de Rham cohomology group. 0→ Ω (M) d −→ Ω (M) d −→ ... d −→ Ω (M) d −→ 0 This is called the de Rham complex. Note that d = 0. If ω = dα, then dω = 0. ...

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 They were generalized by Zeeman in an unpublished work cited in [25] to diffeomorphisms with a hyperbolic chain recurrent set that is axiom-A-diffeomorphisms which satisfy the no cycle condition.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...

Novikov - Morse Theory for Dynamical Systems HuiJun

The present paper contains an interpretation and generalization of Novikov’s theory for Morse type inequalities for closed 1-forms in terms of concepts from Conley’s theory for dynamical systems. We introduce the concept of a flow carrying a cocycle α, (generalized) α-flow for short, where α is a cocycle in bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characteriz...

Holomorphic Morse Inequalities on Covering Manifolds

The goal of this paper is to generalize Demailly’s asymptotic holomorphic Morse inequalities to the case of a covering manifold of a compact manifold. We shall obtain estimates which involve Atiyah’s “normalized dimension” of the square integrable harmonic spaces. The techniques used are those of Shubin who gave a proof for the usual Morse inequalities in the presence of a group action relying ...