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Criteria of measure-preserving for p-adic dynamical systems in terms of the van der Put basis
Linnaeus University, Faculty of Technology, Department of Mathematics.ORCID iD: 0000-0002-9857-0938
Linnaeus University, Faculty of Technology, Department of Mathematics.ORCID iD: 0000-0003-1919-1495
2013 (English)In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 133, no 2, 484-491 p.Article in journal (Refereed) Published
Abstract [en]

This paper is devoted to (discrete) p-adic dynamical systems, an important domain of algebraic and arithmetic dynamics. We consider the following open problem from theory of p-adic dynamical systems. Given continuous function f : Z(p) -> Z(p). Let us represent it via special convergent series, namely van der Put series. How can one specify whether this function is measure-preserving or not for an arbitrary p? In this paper, for any prime p, we present a complete description of all compatible measure-preserving functions in the additive form representation. In addition we prove the criterion in terms of coefficients with respect to the van der Put basis determining whether a compatible function f : Z(p) -> Z(p) preserves the Haar measure. (C) 2012 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
2013. Vol. 133, no 2, 484-491 p.
Keyword [en]
p-Adic numbers, Van der Put basis, Dynamics, Haar measure, Measure-preserving
National Category
Mathematics
Research subject
Mathematics, Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-23526DOI: 10.1016/j.jnt.2012.08.013ISI: 000311769200009OAI: oai:DiVA.org:lnu-23526DiVA: diva2:589442
Available from: 2013-01-18 Created: 2013-01-18 Last updated: 2016-05-03Bibliographically approved
In thesis
1. P-adic dynamical systems and van der Put basis technique
Open this publication in new window or tab >>P-adic dynamical systems and van der Put basis technique
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Växjö: Linnaeus University Press, 2013
Series
Linnaeus University Dissertations, 140/2013
Keyword
dynamical systems, p-adic, 1-Lipschitz, measure-preserving, ergodicity, spheres, uniformly differentiable
National Category
Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-28026 (URN)978-91-87427-37-4 (ISBN)
Public defence
2013-08-27, D1136, Vaxjo, 13:00 (English)
Opponent
Supervisors
Available from: 2013-09-10 Created: 2013-08-10 Last updated: 2015-10-12Bibliographically approved

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CiteExportLink to record
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Citation style
  • apa
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