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On measure-preserving functions over ℤ3.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.ORCID iD: 0000-0003-1919-1495
2012 (English)In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, Vol. 4, no 4, 326-335 p.Article 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. Vol. 4, no 4, 326-335 p.
Keyword [en]
measure-preserving, generalized polynomial, van der Put basis, 3-adic integers
National Category
Mathematics
Research subject
Mathematics, Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-22470DOI: 10.1134/S2070046612040061OAI: oai:DiVA.org:lnu-22470DiVA: diva2:567801
Available from: 2012-11-14 Created: 2012-11-14 Last updated: 2015-10-12Bibliographically 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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