Orbit-counting in non-hyperbolic dynamical systems

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Everest, Graham, Miles, Richard, Stevens, Shaun and Ward, Thomas B. (2007) Orbit-counting in non-hyperbolic dynamical systems. Journal für die reine und angewandte Mathematik (Crelles Journal), 2007 (608). pp. 155-182. ISSN 0075-4102

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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.

Item Type:	Article
Faculty \ School:	Faculty of Science > School of Mathematics (former - to 2024)
UEA Research Groups:	Faculty of Science > Research Groups > Algebra and Combinatorics (former - to 2024) Faculty of Science > Research Groups > Number Theory (former - to 2017) Faculty of Science > Research Groups > Algebra, Logic & Number Theory
Related URLs:	http://www.ams.org/mathscinet-getitem?mr... http://www.zentralblatt-math.org/zmath/e...
Depositing User:	Vishal Gautam
Date Deposited:	18 Mar 2011 14:44
Last Modified:	07 Nov 2024 12:34
URI:	https://ueaeprints.uea.ac.uk/id/eprint/19746
DOI:	10.1515/CRELLE.2007.056

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