Lund, 22nd - 24th January 2004
Quantum Ergodicity - Basic Ideas and Recent Developments
Karl-Olof Lindahl

Ergodicity in p-adic dynamics

Karl-Olof Lindahl (Växjö)

We give necessary and sufficient conditions for ergodicity of dynamics generated by p-adic power series. In [1] it was shown that a p-adic monomial map of the form f(x)=xn is minimal on all spheres of radius less than 1, centered at x=1, if and only if n is a generator of the group of units modulo p2. Moreover, in this case (in fact for all isometries on compact open subsets of the p-adic numbers as shown in [2]), the properties of minimality, ergodicity, and unique ergodicity are equivalent. These results were taken a bit further in [2], for some special type of polynomials. We show that these results are true also for a general power series near a fixed point [3].
[1] M. Gundlach, A. Khrennikov, and K.-O. Lindahl. On ergodic behavior of p-adic dynamical systems. Inf. Dimens. Anal. Quantum Prob. Relat. Top., 4(4):569-577, 2001.
[2] J. Bryk, C.E. Silva. p-adic measurable dynamical systems of simple polynomials. Amer. Math. Monthly, accepted.
[2] K.-O. Lindahl. On conjugation, ergodicity and minimality of p-adic holomorphic functions. Preprint, 2003.

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