S-integer dynamical systems: periodic points.

Ward, T, Everest, G and Chothi, V. (1997) S-integer dynamical systems: periodic points. Journal für die reine und angewandte Mathematik (Crelles Journal), 1997 (489). pp. 99-132. ISSN 0075-4102

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Abstract

We associate via duality a dynamical system to each pair (RS,x), where RS is the ring of S-integers in an A-field k, and x is an element of RS\{0}. These dynamical systems include the circle doubling map, certain solenoidal and toral endomorphisms, full one- and two-sided shifts on prime power alphabets, and certain algebraic cellular automata. In the arithmetic case, we show that for S finite the systems have properties close to hyperbolic systems: the growth rate of periodic points exists and the periodic points are uniformly distributed with respect to Haar measure. The dynamical zeta function is in general irrational however. For S infinite the systems exhibit a wide range of behaviour. Using Heath-Brown's work on the Artin conjecture, we exhibit examples in which S is infinite but the upper growth rate of periodic points is positive.

Item Type:	Article
Faculty \ School:	Faculty of Science > School of Mathematics
Depositing User:	Vishal Gautam
Date Deposited:	18 Mar 2011 14:48
Last Modified:	15 Dec 2022 02:04
URI:	https://ueaeprints.uea.ac.uk/id/eprint/18601
DOI:	10.1515/crll.1997.489.99

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