A Dichotomy in p-adic Dynamics: Measure-preservation of 1-Lipschitz functions vs Bernoullicity of expansive functions

초록

Dynamical systems on the ring Zp of p-adic integers are classi?ed as a dichotomy between 1-Lipschitz functions and expansive functions. In this talk, we formu- late a conjecture for a measure-preservation criterion of 1-Lipschitz functions on Zp, in terms of Mahler?s expansion. Then we verify that it also holds for a wider class of 1-Lipschitz functions that 1-Lipschitz functions that are uniformly di¤erentiable mod p, which is ?rst introduced by Anashin. Also we formulate a conjecture for a Bernoullicity of expansive maps on Zp in Mahler?s expansion and then verify that this conjecture holds for a wider class of expansive maps satisfying certain assumptions. The latter work is based on a Bernoullicity cri- terion of expansive maps on Zp in van der Put?s expansion, which follows from extending the criterion of Khrennikov and Yurova. 1