The rationality of dynamical zeta functions and Woods Hole fixed point formula

TL;DR

This paper proves the rationality of dynamical zeta functions for a broad class of rational functions on the projective line, using Woods Hole fixed point formula and cohomological trace techniques.

Contribution

It establishes the rationality of dynamical zeta functions for many rational functions, extending previous understanding through cohomological methods.

Findings

Dynamical zeta functions are rational for a large class of rational functions.

The proof employs Woods Hole fixed point formula and cohomology trace calculations.

Results connect dynamical systems, algebraic geometry, and fixed point theory.

Abstract

For one variable rational function $\phi\in K(z)$ over a field $K$, we can define a discrete dynamical system by regarding $\phi$ as a self morphism of $\mathbb{P}_{K}^{1}$. Hatjispyros and Vivaldi defined a dynamical zeta function for this dynamical system using multipliers of periodic points, that is, an invariant which indicates the local behavior of dynamical systems. In this paper, we prove the rationality of dynamical zeta functions of this type for a large class of rational functions $\phi\in K(z)$. The proof here relies on Woods Hole fixed point formula and some basic facts on the trace of a linear map acting on cohomology of a coherent sheaf on $\mathbb{P}_{K}^{1}$.