On recent results of ergodic property for p-adic dynamical systems

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• 作者：E. Yurova Axelsson (1)
  
• 关键词：dynamical systems ; p ; adic ; 1 ; Lipschitz ; measure ; preserving ; ergodicity ; spheres ; uniformly differentiable
• 刊名：P-Adic Numbers, Ultrametric Analysis, and Applications
• 出版年：2014
• 出版时间：July 2014
• 年：2014
• 卷：6
• 期：3
• 页码：235-257
• 全文大小：676 KB
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• 作者单位：E. Yurova Axelsson (1)
  
  1. International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science School of Computer Science, Physics and Mathematics, Linnaeus University, 35195, Vaxjo, Sweden
  
• ISSN：2070-0474

文摘

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, 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 paper we present a recent summary of results about 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.