Orbit-counting in non-hyperbolic dynamical systems

G Everest, R Miles, S Stevens, T Ward

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)155-182
Number of pages28
JournalJournal für die reine und angewandte Mathematik (Crelles Journal)
Volume2007
Issue number608
DOIs
Publication statusPublished - 2007

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