S-integer dynamical systems: periodic points.

T Ward, G Everest, V. Chothi

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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.
Original languageEnglish
Pages (from-to)99-132
Number of pages34
JournalJournal für die reine und angewandte Mathematik (Crelles Journal)
Volume1997
Issue number489
DOIs
Publication statusPublished - 1997

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