Related papers: Dynamics of the $p$-adic Shift and Applications

This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let $p$ be a prime number. We first prove the $p$-adic convergence of class…

Monomial mappings, $x\mapsto x^n$, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an anologous result for monomial dynamical systems over $p-$adic…

The purpose of this article is to present, in a simple way, an analytic approach to special numbers and polynomials. The approach is based on the derivative polynomials. The paper is, to some extent, a review article, although it contains…

We conjecture that the structure of Bernoulli numbers can be explicitly given in the closed form $$ B_n = (-1)^{\frac{n}{2}-1} \prod_{p-1 \nmid n} |n|_p^{-1} \prod\limits_{(p,l)\in\Psi^{\rm irr}_1 \atop n \equiv l \mods{p-1}} |p…

Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…

We obtain closed form expressions for convolutions of scale transformations within a certain subset of Appell polynomials. This subset contains the Bernoulli, Apostol-Euler, and Cauchy polynomials, as well as various kinds of their…

In this paper, we study dynamical properties as shadowing and structural stability for a class of dynamics on $\mathbb{Z}_p$ and $\mathbb{Q}_p$, where $p \geq 2$ is a prime number. In particular, we prove that if $f: \mathbb{Z}_p \to…

Let $p$ be an odd prime. For each integer $a$ with $p\nmid a$, the famous Zolotarev's Lemma says that the Legendre symbol $(\frac{a}{p})$ is the sign of the permutation of $\Z/p\Z$ induced by multiplication by $a$. The extension of…

It is significant to study congruences involving multiple harmonic sums. Let $p$ be an odd prime, in recent years, the following curious congruence $$\sum_{\substack{i+j+k=p \\ i, j, k>0}} \frac{1}{i j k} \equiv-2 B_{p-3}\pmod p$$ has been…

Sinai proved that a nonatomic ergodic measure-preserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy and is…

We prove a Closing Lemma for nonuniformly hyperbolic measures of meromorphic maps. We prove also a theorem of approximation of the dynamics of such measures by Bernoulli coding maps.

A permutation p is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as p. The set of realized permutations…

To study any dynamical system it is useful to find a partition that allows essentially faithful encoding (injective, up to a small exceptional set) into a subshift. Most topological and measure-theoretic systems can be represented by…

Given a sequence of independent Bernoulli variables with unknown parameter $p$, and a function $f$ expressed as a power series with non-negative coefficients that sum to at most $1$, an algorithm is presented that produces a Bernoulli…

We study quasi-Monte Carlo integration in a weighted anchored Sobolev space. As the underlying integration nodes we consider Halton sequences in prime bases $\boldsymbol{p}=(p_1,\ldots,p_s)$ which are shifted with a $\boldsymbol{p}$-adic…

A finite abelian $p$-group having an automorphism $x$ such that $1+\ldots+x^{p-1}=0$, can be viewed as a module over an appropriate discrete valuation ring $\mathcal{O}$ containing $\mathbb{Z}_p$ (the ring of $p$-adic integer). This yields…

We show that special perturbations of a particular holomorphic map on $\mathbf{P}^k$ give us examples of maps that possess chaotic nonalgebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we…

Let $p\geq 3$ be a prime. The hyper-algebraic elements in the $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ form an algebraically closed subfield $\mathbb{L}_p^{\operatorname{ha}}$. In this article, we clarify the relations among the fields…

We prove that if $G$ is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of $G$ on ${\underset g \in G \to \Pi} (X_0, \mu_0)_g$ for $(X_{0},\mu_{0})$ an arbitrary…

In this we give a detailed proof of fermionic p-adic q-measures on Z_p and we will treat some interesting formulae related q-extension of Euler numbers and polynomials.