V. Chothi
S-integer dynamical systems: periodic points
Chothi, V.; Everest, G.; Ward, T.
Authors
G. Everest
T. Ward
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 29, 2024 |
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 |
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