Lund, 22^{nd} - 24^{th} January 2004

Quantum Ergodicity - Basic Ideas and Recent Developments

# 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)=x*^{n}
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 *p*^{2}. 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].

References:

[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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