P-adic Dynamical Systems Research Articles

The long-term behavior of the p-adic Sigmoid Beverton-Holt dynamical systems in the projective line [formula omitted

In this paper we study the long-term behavior of the p-adic dynamical systems associated with the Sigmoid Beverton-Holt model that arises in population dynamics. The corresponding fixed points, maximal Siegel disks, attractors, and periodic trajectories are analyzed in the projective line P1(Qp). Moreover, the Julia and Fatou sets associated with these dynamical systems are also examined.

Substitution maps in the Robba ring

We ask several questions about substitution maps in the Robba ring. These questions are motivated by p-adic Hodge theory and the theory of p-adic dynamical systems. We provide answers to those questions in special cases, thereby generalizing results of Kedlaya, Colmez, and others.

On a non-linear p-adic dynamical system

We investigate the behavior of trajectories of a $(3,2)$-rational $p$-adic dynamical system in the complex $p$-adic filed ${\mathbb C}_p$, when there exists a unique fixed point $x_0$. We study this $p$-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point $x_0$). We show that there exists a radius $r$ depending on parameters of the rational function such that: when $x_0$ is an attracting point then the trajectory of an inner point from the ball $U_r(x_0)$ goes to $x_0$ and each sphere with a radius $>r$ (with the center at $x_0$) is invariant; When $x_0$ is a repeller point then the trajectory of an inner point from a ball $U_r(x_0)$ goes forward to the sphere $S_r(x_0)$. Once the trajectory reaches the sphere, in the next step it either goes back to the interior of $U_r(x_0)$ or stays in $S_r(x_0)$ for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of $U_r(x_0)$ it will stay (for all the rest of time) in the sphere (outside of $U_r(x_0)$) that it reached first.

Non-Archimedean Geometry and Applications

The workshop focused on recent developments in non-Archimedean analytic geometry with various applications to other fields. The topics of the talks included applications to complex geometry, mirror symmetry, p-adic Hodge theory, tropical geometry, resolution of singularities, p-adic dynamical systems and diophantine geometry.

P-Adic Dynamical Systems of the Function ax/x2 + a

We show that any (1, 2)-rational function with a unique fixed point is topologically conjugate to a (2, 2)-rational function or to the function f(x) = ax/x2 + a. The case (2, 2) was studied in our previous paper, here we study the dynamical systems generated by the function f on the set of complex p-adic field ℂp. We show that the unique fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We construct the corresponding Siegel disk of these dynamical systems. We determine a sufficiently small set containing the set of limit points. It is given all possible invariant spheres.We show that the p-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure on the set of p-adic numbers ℚp.Moreover some periodic orbits of the system are investigated.

The crystalline period of a height one 𝑝-adic dynamical system

Letffbe a continuous ring endomorphism of\mathbf {Z}_p\lBrack x\rBrack /\mathbf {Z}_pof degreep.p.We prove that ifffacts on the tangent space at00by a uniformizer and commutes with an automorphism of infinite order, then it is necessarily an endomorphism of a formal group overZp.\mathbf {Z}_p.The proof relies on finding a stable embedding of\mathbf {Z}_p\lBrack x\rBrackin Fontaine’s crystalline period ring with the property thatffappears in the monoid of endomorphisms generated by the Galois group ofQp\mathbf {Q}_pand crystalline Frobenius. Our result verifies, overZp,\mathbf {Z}_p,the height one case of a conjecture by Lubin.

P-adic dynamical systems of (2,2)-rational functions with unique fixed point

We consider a family of (2, 2)-rational functions given on the set of complex p-adic field Cp. Each such function has a unique fixed point. We study p-adic dynamical systems generated by the (2, 2)-rational functions. We show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. Siegel disks of these dynamical systems are found. We obtain an upper bound for the set of limit points of each trajectory, i.e., we determine a sufficiently small set containing the set of limit points. For each (2, 2)-rational function on Cp there are two points x^1=x^1(f),x^2=x^2(f)∈Cp which are zeros of its denominator. We give explicit formulas of radiuses of spheres (with the center at the fixed point) containing some points such that the trajectories (under actions of f) of the points after a finite step come to x^1 or x^2. Moreover for a class of (2, 2)-rational functions we study ergodicity properties of the dynamical systems on the set of p-adic numbers Qp. For each such function we describe all possible invariant spheres. We show that if p ≥ 3 then the p-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure. In case p=2 under some conditions we prove non ergodicity and we show that there exists a sphere on which our dynamical system is ergodic.

Chaotic behavior of the <i>P</i>-adic Potts-Bethe mapping

In the previous investigations of the authors the renormalization group method to p-adic models on Cayley trees has been developed. This method is closely related to the investigation of p-adic dynamical systems associated with a given model. In this paper, we study chaotic behavior of the Potts-Bethe mapping. We point out that a similar kind of result is not known in the case of real numbers (with rigorous proofs).

Erratum to: “p-adic dynamical systems of Chebyshev polynomials”

Erratum to: “p-adic dynamical systems of Chebyshev polynomials”

P-Adic (3, 2)-rational dynamical systems

We investigate behavior of trajectory of a (3, 2)-rational p-adic dynamical system in complex p-adic field \(\mathbb{C}_p\). The paper studies Siegel disks and attractors of these dynamical systems. The set of fixed points of the (3, 2)-rational function may by empty, or may consist of a single element, or of two elements. We obtained the following results. In the case of existence of two fixed points, the p-adic dynamical system has a very rich behavior: we show that Siegel disks may either coincide or be disjoint for different fixed points of the dynamical system. Besides, we find the basin of the attractor of the system. For some values of the parameters there are trajectories which go arbitrary far from the fixed points.

On recent results of ergodic property for p-adic dynamical systems

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc. In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions. In this paper we present a recent summary of results about the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions. The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball. Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.

Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions

This paper is devoted to the problem of ergodicity of p-adic dynamical systems. We solved the problem of characterization of ergodicity and measure preserving for (discrete) p-adic dynamical systems for arbitrary prime p for iterations based on 1-Lipschitz functions. This problem was open since long time and only the case p=2 was investigated in details. We formulated the criteria of ergodicity and measure preserving in terms of coordinate functions corresponding to digits in the canonical expansion of p-adic numbers. (The coordinate representation can be useful, e.g., for applications to cryptography.) Moreover, by using this representation we can consider non-smooth p-adic transformations. The basic technical tools are van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theorems by presenting concrete classes of ergodic functions. As is well known, p-adic spaces have the fractal (although very special) structure. Hence, our study covers a large class of dynamical systems on fractals. Dynamical systems under investigation combine simplicity of the algebraic dynamical structure with very high complexity of behavior.

Qualitative theory of p-adic dynamical systems

We consider a class of dynamical systems over the p-adic number field: hierarchical dynamical systems. We prove a strong variant of the Poincare theorem on the number of returns for such systems and show that hierarchical systems do not admit mixing. We describe hierarchical dynamical systems over the projective line and present an example of a nonhierarchical p-adic system that admits mixing: the p-adic baker’s transformation.

On a nonlinear p-adic dynamical system

We investigate the behavior of trajectories of a (3, 2)-rational p-adic dynamical system in the complex p-adic field ℂ p , when there exists a unique fixed point x 0. We study this p-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point x 0). We show that there exists a radius r depending on parameters of the rational function such that: when x 0 is an attracting point then the trajectory of an inner point from the ball U r (x 0) goes to x 0 and each sphere with a radius > r (with the center at x 0) is invariant; When x 0 is a repeller point then the trajectory of an inner point from a ball U r (x 0) goes forward to the sphere S r (x 0). Once the trajectory reaches the sphere, in the next step it either goes back to the interior of U r (x 0) or stays in S r (x 0) for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of U r(x 0) it will stay (for all the rest of time) in the sphere (outside of U r(x 0)) that it reached first.

P-Adic dynamical systems of Chebyshev polynomials

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ℚp, enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of Tp: they are simple and the nonzero ones lie in the unit circle of the unramified extension of ℚp, (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of Tp in the field ℚp, (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of Tn is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.

On ergodicity of p-adic dynamical systems for arbitrary prime p

In this paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of 1-Lipschitz p-adic functions that are defined on (and valuated in) the space ℤp of p-adic integers for any prime p. The conditions are stated in terms of coordinate representations of p-adic functions.

On a p-adic Cubic Generalized Logistic Dynamical System

Applications of p-adic numbers mathematical physics, quantum mechanics stimulated increasing interest in the study of p-adic dynamical system. One of the interesting investigations is p-adic logistics map. In this paper, we consider a new generalization, namely we study a dynamical system of the form fa(x) = ax(1−x2). The paper is devoted to the investigation of a trajectory of the given system. We investigate the generalized logistic dynamical system with respect to parameter a and we restrict ourselves for the investigation of the case |a|p < 1. We study the existence of the fixed points and their behavior. Moreover, we describe their size of attractors and Siegel discs since the structure of the orbits of the system is related to the geometry of the p-adic Siegel discs.

A Dynamical System Approach to Phase Transitions for p-Adic Potts Model on the Cayley Tree of Order Two

In the present paper, we introduce a new kind of p-adic measures for (q + 1)-state Potts model, called generalized p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. We employ a dynamical system approach to establish phase transition phenomena for the given model. Namely, using the derived recursive relations we define a one-dimensional fractional p-adic dynamical system. We show that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. In this case, there exists a strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields the existence of a quasi phase transition.

Ergodicity of dynamical systems on 2-adic spheres

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, e.g., in physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc.In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions.In this thesis we restrict our study to the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions.The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball.Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.In this thesis, we present a description of 1-Lipschitz measure-preserving and ergodic functions for arbitrary prime p.

On measure-preserving functions over ℤ3

This paper is devoted to (discrete) p-adic dynamical systems, an important domain ofalgebraic and arithmetic dynamics [31]-[41], [5]-[8]. In this note we study properties of measurepreservingdynami ...