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P-adic dynamical systems and van der Put basis technique
Linnaeus University, Faculty of Technology, Department of Mathematics.ORCID iD: 0000-0003-1919-1495
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
Keywords [en]
dynamical systems, p-adic, 1-Lipschitz, measure-preserving, ergodicity, spheres, uniformly differentiable
National Category
Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-28026ISBN: 978-91-87427-37-4 (print)OAI: oai:DiVA.org:lnu-28026DiVA, id: diva2:639865
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
List of papers
1. Using van der Put basis to determine if a 2-adic function is measure-preserving or ergodic w.r.t. Haar measure
Open this publication in new window or tab >>Using van der Put basis to determine if a 2-adic function is measure-preserving or ergodic w.r.t. Haar measure
2011 (English)In: Advances in Non-Archimedean Analysis: 11th International Conference p-adic Funcional Analysis, July 5-9, 2010, Université Blaise Pascal, Clemont-Ferrand, France / [ed] Jesus Araujo-Gomez, Bertin Diarra and Alain Escassut, American Mathematical Society (AMS), 2011, Vol. 551, p. 33-38Chapter in book (Refereed)
Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2011
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-16578 (URN)10.1090/conm/551 (DOI)978-0-8218-5291-0 (p) (ISBN)978-0-8218-8230-6 (e) (ISBN)
Available from: 2012-01-04 Created: 2012-01-04 Last updated: 2016-05-03Bibliographically approved
2. T-functions revisited: new criteria for bijectivity/transitivity
Open this publication in new window or tab >>T-functions revisited: new criteria for bijectivity/transitivity
2014 (English)In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 71, no 3, p. 383-407Article in journal (Refereed) Published
Abstract [en]

The paper presents new criteria for bijectivity/transitivity of T-functions and a fast knapsack-like algorithm of evaluation of a T-function. Our approach is based on non-Archimedean ergodic theory: Both the criteria and algorithm use van der Put series to represent 1-Lipschitz p-adic functions and to study measure-preservation/ergodicity of these.

Place, publisher, year, edition, pages
Springer Netherlands, 2014
Keywords
T-function – Bijectivity – Transitivity – Non-Archimedean ergodic theory – van der Put series – Ergodicity – Measure-preservation
National Category
Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-21818 (URN)10.1007/s10623-012-9741-z (DOI)000334179100002 ()2-s2.0-84899125882 (Scopus ID)
Available from: 2012-09-26 Created: 2012-09-26 Last updated: 2017-12-07Bibliographically approved
3. Characterization of ergodicity of p-adic dynamical systems by using the van der Put basis.
Open this publication in new window or tab >>Characterization of ergodicity of p-adic dynamical systems by using the van der Put basis.
2011 (English)In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 83, no 3, p. 306-308Article in journal (Refereed) Published
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-16579 (URN)10.1134/S1064562411030100 (DOI)2-s2.0-80052689155 (Scopus ID)
Available from: 2012-01-04 Created: 2012-01-04 Last updated: 2017-12-08Bibliographically approved
4. Ergodicity of dynamical systems on 2-adic spheres
Open this publication in new window or tab >>Ergodicity of dynamical systems on 2-adic spheres
2012 (English)In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 86, no 3, p. 843-845Article in journal (Refereed) Published
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-24550 (URN)10.1134/S1064562412060312 (DOI)000313201600031 ()2-s2.0-84893469846 (Scopus ID)
Available from: 2013-02-25 Created: 2013-02-25 Last updated: 2017-12-06Bibliographically approved
5. Criteria of measure-preserving for p-adic dynamical systems in terms of the van der Put basis
Open this publication in new window or tab >>Criteria of measure-preserving for p-adic dynamical systems in terms of the van der Put basis
2013 (English)In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 133, no 2, p. 484-491Article 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.

Keywords
p-Adic numbers, Van der Put basis, Dynamics, Haar measure, Measure-preserving
National Category
Mathematics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-23526 (URN)10.1016/j.jnt.2012.08.013 (DOI)000311769200009 ()2-s2.0-84867664993 (Scopus ID)
Available from: 2013-01-18 Created: 2013-01-18 Last updated: 2017-12-06Bibliographically approved
6. On measure-preserving functions over ℤ3.
Open this publication in new window or tab >>On measure-preserving functions over ℤ3.
2012 (English)In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 4, no 4, p. 326-335Article in journal (Refereed) Published
Abstract [en]

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 measurepreservingdynamical systems in the case p = 3. This case differs crucially from the case p = 2.The latter was studied in the very detail in [43]. We state results on all compatible functions whichpreserve measure on the space of 3-adic integers, using previous work of A. Khrennikov and authorof present paper, see [24]. To illustrate one of the obtained theorems we describe conditions for the3-adic generalized polynomial to be measure-preserving on Z3. The generalized polynomials withintegral coefficients were studied in [17, 33] and represent an important class of T-functions. Inturn, it is well known that T-functions are well-used to create secure and efficient stream ciphers,pseudorandom number generators.

Place, publisher, year, edition, pages
Maik Nauka/Interperiodica, 2012
Keywords
measure-preserving, generalized polynomial, van der Put basis, 3-adic integers
National Category
Mathematics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-22470 (URN)10.1134/S2070046612040061 (DOI)2-s2.0-84966989689 (Scopus ID)
Available from: 2012-11-14 Created: 2012-11-14 Last updated: 2017-12-07Bibliographically approved
7. Study of ergodicity of p-adic dynamical systems with the aid of van der Put basis
Open this publication in new window or tab >>Study of ergodicity of p-adic dynamical systems with the aid of van der Put basis
2011 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The study of p-adic dynamical systems is motivated by their applications in various (and surprisingly diverse) areas of mathematics, e.g., in physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc.

In this thesis we use decomposition of a continuous function f : zp -> zp into a convergent van der Put series to determine whether f is 1-Lipschitz, measure-preserving and/or ergodic.

The main mathematical tool used in this research is the representation of the function by the van der Put series, which are special convergent series from p-adic analysis.

This is the first attempt to use the van der Put basis to examine the properties of (discrete) dynamical systems in fields of p-adic numbers. Note that the van der Put basis differs fundamentally from previously used ones, for example, the monomial and Mahler bases, which are related to the algebraic structure of p-adic fields.

The van der Put basis is related to the zero dimensional topology of these fields (ultrametric structure), since it consists of characteristic functions of p-adic balls; i.e., the basic point in the construction of this basis is the continuity of the characteristic function of a p-adic ball.

Place, publisher, year, edition, pages
Växjö: Linnaeus University, 2011
Keywords
p-adic numbers, van der Put basis, 1-Lipschitz, measure-preserving, ergodicity, sphere, T-function
National Category
Other Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-15821 (URN)
Presentation
2011-09-27, 13:15 (English)
Opponent
Supervisors
Funder
Swedish Research Council
Available from: 2012-01-30 Created: 2011-11-15 Last updated: 2017-09-01Bibliographically approved
8. Van der Put basis and p-adic dynamics
Open this publication in new window or tab >>Van der Put basis and p-adic dynamics
2010 (English)In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 2, no 2, p. 175-178Article in journal (Other academic) Published
Place, publisher, year, edition, pages
MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC., 2010
Keywords
van der Put basis, p-adic dynamical systems, ergodicity
National Category
Other Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-9991 (URN)10.1134/S207004661002007X (DOI)
Available from: 2011-01-14 Created: 2011-01-14 Last updated: 2017-12-11Bibliographically approved

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Yurova Axelsson, Ekaterina

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