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P-adic dynamical systems and van der Put basis techniquePrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2013 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### 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

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt513",{id:"formSmash:j_idt513",widgetVar:"widget_formSmash_j_idt513",multiple:true}); Available from: 2013-09-10 Created: 2013-08-10 Last updated: 2015-10-12Bibliographically approved
##### List of papers

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.

1. Using van der Put basis to determine if a 2-adic function is measure-preserving or ergodic w.r.t. Haar measure$(function(){PrimeFaces.cw("OverlayPanel","overlay472854",{id:"formSmash:j_idt562:0:j_idt567",widgetVar:"overlay472854",target:"formSmash:j_idt562:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. T-functions revisited: new criteria for bijectivity/transitivity$(function(){PrimeFaces.cw("OverlayPanel","overlay557062",{id:"formSmash:j_idt562:1:j_idt567",widgetVar:"overlay557062",target:"formSmash:j_idt562:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Characterization of ergodicity of p-adic dynamical systems by using the van der Put basis.$(function(){PrimeFaces.cw("OverlayPanel","overlay472858",{id:"formSmash:j_idt562:2:j_idt567",widgetVar:"overlay472858",target:"formSmash:j_idt562:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Ergodicity of dynamical systems on 2-adic spheres$(function(){PrimeFaces.cw("OverlayPanel","overlay607632",{id:"formSmash:j_idt562:3:j_idt567",widgetVar:"overlay607632",target:"formSmash:j_idt562:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Criteria of measure-preserving for p-adic dynamical systems in terms of the van der Put basis$(function(){PrimeFaces.cw("OverlayPanel","overlay589442",{id:"formSmash:j_idt562:4:j_idt567",widgetVar:"overlay589442",target:"formSmash:j_idt562:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. On measure-preserving functions over ℤ_{3.}$(function(){PrimeFaces.cw("OverlayPanel","overlay567801",{id:"formSmash:j_idt562:5:j_idt567",widgetVar:"overlay567801",target:"formSmash:j_idt562:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

7. Study of ergodicity of *p*-adic dynamical systems with the aid of van der Put basis$(function(){PrimeFaces.cw("OverlayPanel","overlay456780",{id:"formSmash:j_idt562:6:j_idt567",widgetVar:"overlay456780",target:"formSmash:j_idt562:6:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

8. Van der Put basis and p-adic dynamics$(function(){PrimeFaces.cw("OverlayPanel","overlay387724",{id:"formSmash:j_idt562:7:j_idt567",widgetVar:"overlay387724",target:"formSmash:j_idt562:7:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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