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Study of ergodicity of p-adic dynamical systems with the aid of van der Put basis
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.ORCID iD: 0000-0003-1919-1495
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 [en]
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: urn:nbn:se:lnu:diva-15821Libris ID: 12516256OAI: oai:DiVA.org:lnu-15821DiVA, id: diva2:456780
Presentation
2011-09-27, 13:15 (English)
Opponent
Supervisors
Funder
Swedish Research CouncilAvailable from: 2012-01-30 Created: 2011-11-15 Last updated: 2017-09-01Bibliographically 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
Keywords
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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Yurova, Ekaterina

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