Orbit-counting in non-hyperbolic dynamical systems

EVEREST, Graham, MILES, Richard, STEVENS, Shaun and WARD, Thomas (2007). Orbit-counting in non-hyperbolic dynamical systems. Journal für die reine und angewandte Mathematik, 608, 155-182.

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Official URL: https://doi.org/10.1515/CRELLE.2007.056
Link to published version:: https://doi.org/10.1515/CRELLE.2007.056

Abstract

There are well-known analogues 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. The proof of the dynamical analogue of Mertens' Theorem uses transcendence theory and Dirichlet characters.

Item Type: Article
Departments: Faculty of Social Sciences and Humanities > Teacher Education
Identification Number: https://doi.org/10.1515/CRELLE.2007.056
Depositing User: Richard Miles
Date Deposited: 26 Jan 2018 13:35
Last Modified: 26 Jan 2018 13:35
URI: http://shura.shu.ac.uk/id/eprint/17237

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