Level sets of multiple ergodic averages

We propose to study multiple ergodic averages from multifractal analysis point of view. In some special cases in the symbolic dynamics, Hausdorff dimensions of the level sets of multiple ergodic average limit are determined by using Riesz products.Comment: This note was refused by Proceedings of AMS although the referee said "In my opinion this is a nice application of the Riesz product technique to solve, in principle, a hard problem when considered in its full generality. Nevertheless, I think it needs some extra work to see how this example seats in a more general context and explore how far this technique can go." We should say that Riesz product works perfectly in the situation described in this note, but Riesz product has its limit--we don't think that Riesz product technique can solve the problem in its generalit

On minimal decomposition of pp-adic polynomial dynamical systems

A polynomial of degree $\ge 2$ with coefficients in the ring of $p$-adic numbers $\mathbb{Z}_p$ is studied as a dynamical system on $\mathbb{Z}_p$. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on $\mathbb{Z}_2$, we exhibit all its minimal subsystems.Comment: 27 page

Generic points in systems of specification and Banach valued Birkhoff ergodic average

We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measure-theoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variational principle for its topological entropy spectrum. As application, we examine a particular example concerning with the set of real numbers for which the frequencies of occurrences in their dyadic expansions of infinitely many words are prescribed. This relies on our explicit determination of a maximal entropy measure.Comment: accepted by Discrete and Continuous Dynamical System