Orbit-counting in non-hyperbolic dynamical systems
Dynamical Systems
2007-09-20 v1 Number Theory
Abstract
There are well-known analogs of the prime number theorem and Mertens' theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbit-counting function. Mertens' Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the non-hyperbolic eigendirections.
Cite
@article{arxiv.math/0511569,
title = {Orbit-counting in non-hyperbolic dynamical systems},
author = {G. Everest and R. Miles and S. Stevens and T. Ward},
journal= {arXiv preprint arXiv:math/0511569},
year = {2007}
}