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 |
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Faculty \ School: | Faculty of Science > School of Mathematics |

Related URLs: | |

Depositing User: | Vishal Gautam |

Date Deposited: | 18 Mar 2011 14:48 |

Last Modified: | 03 Oct 2022 04:37 |

URI: | https://ueaeprints.uea.ac.uk/id/eprint/18601 |

DOI: | 10.1515/crll.1997.489.99 |

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