377 research outputs found

### On minimal decomposition of $p$-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

### 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

### Minimality of p-adic rational maps with good reduction

A rational map with good reduction in the field $\mathbb{Q}\_p$ of $p$-adic
numbers defines a $1$-Lipschitz dynamical system on the projective line
$\mathbb{P}^1(\mathbb{Q}\_p)$ over $\mathbb{Q}\_p$. The dynamical structure of
such a system is completely described by a minimal decomposition. That is to
say, $\mathbb{P}^1(\mathbb{Q}\_p)$ is decomposed into three parts: finitely
many periodic orbits; finite or countably many minimal subsystems each
consisting of a finite union of balls; and the attracting basins of periodic
orbits and minimal subsystems. For any prime $p$, a criterion of minimality for
rational maps with good reduction is obtained. When $p=2$, a condition in terms
of the coefficients of the rational map is proved to be necessary for the map
being minimal and having good reduction, and sufficient for the map being
minimal and $1$-Lipschitz. It is also proved that a rational map having good
reduction of degree $2$, $3$ and $4$ can never be minimal on the whole space
$\mathbb{P}^1(\mathbb{Q}\_2)$.Comment: 21 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

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