Orbits for products of maps

TL;DR

This paper investigates the properties of dynamical zeta functions and orbit Dirichlet series for product maps, revealing how their convergence characteristics behave and identifying a natural boundary in specific cases.

Contribution

It provides new insights into the convergence behavior of these series for product maps and demonstrates the existence of a natural boundary in the orbit Dirichlet series of certain maps.

Findings

The radius of convergence for the zeta function under products is characterized.

The abscissa of convergence for the orbit Dirichlet series is analyzed for product maps.

The orbit Dirichlet series of the cube of a map with one orbit per length has a natural boundary.

Abstract

We study the behaviour of the dynamical zeta function and the orbit Dirichlet series for products of maps. The behaviour under products of the radius of convergence for the zeta function, and the abscissa of convergence for the orbit Dirichlet series, are discussed. The orbit Dirichlet series of the cartesian cube of a map with one orbit of each length is shown to have a natural boundary.