S-integer dynamical systems: periodic points

Chothi, V.; Everest, G.; Ward, T.

Abstract

We associate via duality a dynamical system to each pair (R_S,x), where R_S is the ring of S-integers in an A-field k, and x is an element of R_S\{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.

Citation

Chothi, V., Everest, G., & Ward, T. (1997). S-integer dynamical systems: periodic points. Journal für die reine und angewandte Mathematik, 1997(489), 99-132. https://doi.org/10.1515/crll.1997.489.99

• Journal Article Type: Article
• Publication Date: Jan 1, 1997
• Deposit Date: Oct 12, 2012
• Publicly Available Date: Mar 18, 2014
• Journal: Journal für die reine und angewandte Mathematik
• Print ISSN: 0075-4102
• Electronic ISSN: 1435-5345
• Publisher: De Gruyter
• Peer Reviewed: Peer Reviewed
• Volume: 1997
• Issue: 489
• Pages: 99-132
• DOI: https://doi.org/10.1515/crll.1997.489.99
• Public URL: https://durham-repository.worktribe.com/output/1502386