Mertens' theorem for toral automorphisms

TL;DR

This paper establishes a dynamical version of Mertens' theorem for ergodic toral automorphisms, analyzing the impact of eigenvalue resonances on the distribution of periodic orbits and providing examples with varying orbit counts.

Contribution

It introduces a Mertens' theorem for toral automorphisms with an explicit error term and explores how eigenvalue resonances affect periodic orbit counts.

Findings

Error term of O(N^{-1}) in the theorem

Resonances among eigenvalues influence orbit distribution

Examples with significantly more or fewer orbits than expected

Abstract

A dynamical Mertens' theorem for ergodic toral automorphisms with error term O(N^{-1}) is found, and the influence of resonances among the eigenvalues of unit modulus is examined. Examples are found with many more, and with many fewer, periodic orbits than expected.