377 research outputs found

    On minimal decomposition of pp-adic polynomial dynamical systems

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    A polynomial of degree ≥2\ge 2 with coefficients in the ring of pp-adic numbers Zp\mathbb{Z}_p is studied as a dynamical system on Zp\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 Z2\mathbb{Z}_2, we exhibit all its minimal subsystems.Comment: 27 page

    Level sets of multiple ergodic averages

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    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

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    A rational map with good reduction in the field Q_p\mathbb{Q}\_p of pp-adic numbers defines a 11-Lipschitz dynamical system on the projective line P1(Q_p)\mathbb{P}^1(\mathbb{Q}\_p) over Q_p\mathbb{Q}\_p. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, P1(Q_p)\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 pp, a criterion of minimality for rational maps with good reduction is obtained. When p=2p=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 11-Lipschitz. It is also proved that a rational map having good reduction of degree 22, 33 and 44 can never be minimal on the whole space P1(Q_2)\mathbb{P}^1(\mathbb{Q}\_2).Comment: 21 page

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

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    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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