Uniqueness of dynamical zeta functions and symmetric products

TL;DR

This paper characterizes a class of dynamical zeta functions through axioms, proving the Lefschetz zeta function's uniqueness and exploring its connection to symmetric products with various applications.

Contribution

It introduces axioms for dynamical zeta functions, establishes the Lefschetz zeta function's uniqueness, and links it to symmetric products for new insights.

Findings

Lefschetz zeta function is uniquely characterized among dynamical zeta functions.

Axioms extending topological degree characterize these zeta functions.

Applications derived from symmetric product interpretations.

Abstract

A characterization of dynamically defined zeta functions is presented. It comprises a list of axioms, natural extension of the one which characterizes topological degree, and a uniqueness theorem. Lefschetz zeta function is the main (and proved unique) example of such zeta functions. Another interpretation of this function arises from the notion of symmetric product from which some corollaries and applications are obtained.