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S-integer dynamical systems : periodic points.

Chothi, V. and Everest, G. and Ward, T. (1997) 'S-integer dynamical systems : periodic points.', Journal für die reine und angewandte Mathematik. = Crelles journal., 1997 (489). pp. 99-132.


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.

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Last Modified:18 Mar 2014 16:45

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