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Study of ergodicity of p-adic dynamical systems with the aid of van der Put basis
Linnéuniversitetet, Fakultetsnämnden för naturvetenskap och teknik, Institutionen för datavetenskap, fysik och matematik, DFM.ORCID-id: 0000-0003-1919-1495
2011 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Växjö: Linnaeus University , 2011.
Nyckelord [en]
p-adic numbers, van der Put basis, 1-Lipschitz, measure-preserving, ergodicity, sphere, T-function
Nationell ämneskategori
Annan matematik
Forskningsämne
Matematik, Tillämpad matematik
Identifikatorer
URN: urn:nbn:se:lnu:diva-15821Libris ID: 12516256OAI: oai:DiVA.org:lnu-15821DiVA, id: diva2:456780
Presentation
2011-09-27, 13:15 (Engelska)
Opponent
Handledare
Forskningsfinansiär
VetenskapsrådetTillgänglig från: 2012-01-30 Skapad: 2011-11-15 Senast uppdaterad: 2017-09-01Bibliografiskt granskad
Ingår i avhandling
1. P-adic dynamical systems and van der Put basis technique
Öppna denna publikation i ny flik eller fönster >>P-adic dynamical systems and van der Put basis technique
2013 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Växjö: Linnaeus University Press, 2013
Serie
Linnaeus University Dissertations ; 140
Nyckelord
dynamical systems, p-adic, 1-Lipschitz, measure-preserving, ergodicity, spheres, uniformly differentiable
Nationell ämneskategori
Matematik
Forskningsämne
Matematik, Tillämpad matematik
Identifikatorer
urn:nbn:se:lnu:diva-28026 (URN)9789187427374 (ISBN)
Disputation
2013-08-27, D1136, Vaxjo, 13:00 (Engelska)
Opponent
Handledare
Tillgänglig från: 2013-09-10 Skapad: 2013-08-10 Senast uppdaterad: 2024-02-06Bibliografiskt granskad

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Yurova, Ekaterina

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