Resolvent of vector fields and Lefschetz numbers

TL;DR

This paper establishes a general relation between dynamical series and the resolvent of vector fields, extending to all smooth flows, and introduces new series linked to topological invariants.

Contribution

It provides a universal formula connecting dynamical series with the resolvent kernel for any smooth flow, not just hyperbolic ones.

Findings

Derived a general formula relating dynamical series to the resolvent kernel.

Introduced new dynamical series and computed their values at zero.

Linked these series to topological invariants.

Abstract

Dynamical series such as the Ruelle zeta function have become a staple in the study of hyperbolic flows. They are usually analyzed by relating them to the resolvent of the vector field. In this paper we give the general form of such relations, which involves the intersection of the kernel of said resolvent with integration currents. Our formula is actually valid for any smooth flow, not necessarily hyperbolic. As an application, we introduce certain dynamical series that had not appeared before. Finally, we compute their value at zero, and their relation with topological invariants.