Orbit counting with an isometric direction

TL;DR

This paper investigates a 3-adic extension of the circle doubling map, focusing on orbit counting in an isometric direction, revealing weaker asymptotic results due to loss of hyperbolicity.

Contribution

It introduces a 3-adic extension with an isometric eigendirection, extending orbit counting results to non-hyperbolic dynamical systems.

Findings

Weaker asymptotic orbit counting results compared to hyperbolic systems

Identification of 3-adic eigendirection with isometric behavior

Extension of prime number theorem analogues to non-hyperbolic maps

Abstract

Analogues of the prime number theorem and Merten's theorem are well-known for dynamical systems with hyperbolic behaviour. In this paper a 3-adic extension of the circle doubling map is studied. The map has a 3-adic eigendirection in which it behaves like an isometry, and the loss of hyperbolicity leads to weaker asymptotic results on orbit counting than those obtained for hyperbolic maps.