Minimal polynomial dynamics on the set of 3-adic integers

p-adic根分离与整数多项式的判别式

p-adic root separation and the discriminant of integer polynomials

研究固定次数和有界高度的整数多项式的p-adic根分离问题，以及判别式被素数幂整除的计数问题，证明了存在大量具有几乎素数幂判别式的不可约多项式。

Ergodic Transformations of the Space of $p$-adic Integers

Let $\mathcal L_1$ be the set of all mappings $f\colon\Z_p\Z_p$ of the space of all $p$-adic integers $\Z_p$ into itself that satisfy Lipschitz condition with a constant 1. We prove that the mapping $f\in\mathcal L_1$ is ergodic with respect to the normalized Haar measure on $\Z_p$ if and only if $f$ induces a single cycle permutation on each residue ring $\Z/p^k\Z$ modulo $p^k$, for all $k=1,2,3,...$. The multivariate case, as well as measure-preserving mappings, are considered also. Results of the paper in a combination with earlier results of the author give explicit description of ergodic …

Dynamic Structures of 2-adic Fibonacci Polynomials

We prove recurrence relations and modulo periodic properties of multiple derivatives of Fibonacci polynomials. We apply the obtained results to present the dynamic structures of Fibonacci polynomials over the ring of 2-adic integers by investigating minimal decompositions which consist of minimal subsystems and attracting basins.

Minimality of $5$-adic polynomial dynamical systems

We characterize the dynamical systems consisting of the set of 5-adic integers and polynomial maps which consist of only one minimal component.

An introduction to $p$-adic systems: A new kind of number systems inspired by the Collatz $3n+1$ conjecture

This article introduces a new kind of number systems on $p$-adic integers which is inspired by the well-known $3n+1$ conjecture of Lothar Collatz. A $p$-adic system is a piecewise function on $\mathbb{Z}_p$ which has branches for all residue classes modulo $p$ and whose dynamics can be used to define digit expansions of $p$-adic integers which respect congruency modulo powers of $p$ and admit a distinctive "block structure". $p$-adic systems generalize several notions related to $p$-adic integers such as permutation polynomials and put them under a common framework, allowing for results and te…

整数与 $p$-进整数的计数和 Hausdorff 测度

Counting and Hausdorff measures for integers and $p$-adic integers

本文利用 $p$-进整数的分形结构研究整数集合，建立计数测度界、计数维数与盒计数维数关系，并给出整数集为闭集投影的充要组合条件。

On the automorphism group of minimal S-adic subshifts of finite alphabet rank

It has been recently proved that the automorphism group of a minimal subshift with non-superlinear word complexity is virtually $\mathbb{Z}$ [DDPM15, CK15]. In this article we extend this result to a broader class proving that the automorphism group of a minimal S-adic subshift of finite alphabet rank is virtually $\mathbb{Z}$. The proof is based on a fine combinatorial analysis of the asymptotic classes in this type of subshifts, which we prove are a finite number.

On minimal decomposition of $p$-adic homographic dynamical systems

A homographic map in the field of $p$-adic numbers $\mathbb{Q}_p}$ is studied as a dynamical system on $\mathbb{P}^{1}(\mathbb{Q}_p)$, the projective line over $\mathbb{Q}_p$. If such a system admits one or two fixed points in $\mathbb{Q}_p$, then it is conjugate to an affine dynamics whose dynamical structure has been investigated by Fan and Fares. In this paper, we shall mainly solve the remaining case that the system admits no fixed point. We shall prove that this system can be decomposed into a finite number of minimal subsystems which are topologically conjugate to each other. All the min…

$\mathcal{S}$-adic characterization of minimal ternary dendric shifts

Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux-Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive $\mathcal{S}$-adic representation where the morphisms in $\mathcal{S}$ are positive tame automorphisms of the free group generated by the alphabet. In this paper we investigate those $\mathcal{S}$-adic representations, heading towards an $\mathcal{S}$-adic characterization of this fa…

Uniformly distributed sequences of p-adic integers, II

The paper describes ergodic (with respect to the Haar measure) functions in the class of all functions, which are defined on (and take values in) the ring of p-adic integers, and which satisfy (at least, locally) Lipschitz condition with coefficient 1. Equiprobable (in particular, measure-preserving) functions of this class are described also. In some cases (and especially for p=2) the descriptions are given by explicit formulae. Some of the results may be viewed as descriptions of ergodic isometric dynamical systems on p-adic unit disk. The study was motivated by the problem of pseudorandom n…

Ergodic dynamical systems over the Cartesian power of the ring of p-adic integers

For any 1-lipschitz ergodic map $F:\; \mathbb{Z}^{k}_{p} \mapsto \mathbb{Z}^{k}_{p},\;k>1\in\mathbb{N},$ there are 1-lipschitz ergodic map $G:\; \mathbb{Z}_{p} \mapsto \mathbb{Z}_{p}$ and two bijection $H_k$, $T_{k,\;P}$ that $$G = H_{k} \circ T_{k,\;P}\circ F\circ H^{-1}_{k} \text{ and } F = H^{-1}_{k} \circ T_{k,\;P^{-1}}\circ G\circ H_{k}.$$

Interplay between finite topological rank minimal Cantor systems, $\mathcal S$-adic subshifts and their complexity

Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable ${\mathcal S}$-adic subshifts. This is done establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like all…

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 minimal…

极小树状移位的$\mathcal{S}$进刻画

$\mathcal{S}$-adic characterization of minimal dendric shifts

研究极小树状移位的刻画问题，利用两个有限图给出其$\mathcal{S}$进表示的判定特征，并据此可判定一致循环态射词生成的移位空间是否（最终）为树状。

通过S-adic表示研究极小移位的连续特征值

Continuous eigenvalues of minimal subshifts via S-adic representations and coboundaries

本文通过S-adic表示研究极小移位的连续特征值，结合组合学与线性代数方法，提出特征化条件及非平凡上界不存在的充分条件，并恢复Tijdeman关于转移移位的最小因子复杂度结果。

Dynamics of the $p$-adic Shift and Applications

We present a novel way of realizing the Bernoulli shift on $p$ symbols on the $p$-adic integers, where $p$ is a prime. By showing that suitably small perturbations of the shift are still Bernoulli we find many "nice" maps, such as polynomials on $\mathbb Z_p$, that are isomorphic to the (noninvertible) Bernoulli shift.