Remetrizing dynamical systems to control distances of points in time

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-05-06 DOI:10.1016/j.topol.2025.109419

Krzysztof Gołębiowski

{"title":"Remetrizing dynamical systems to control distances of points in time","authors":"Krzysztof Gołębiowski","doi":"10.1016/j.topol.2025.109419","DOIUrl":null,"url":null,"abstract":"<div><div>Main aim of this article is to prove that for any continuous function <math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></math>, where X is metrizable (or, more generally, for any family <math><mi>F</mi></math> of such functions, with one extra condition), there exists a compatible metric d on X such that the nth iteration of f (more generally, composition of any n functions from <math><mi>F</mi></math>) is Lipschitz with constant <math><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> where <math><msubsup><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math> is an arbitrarily fixed sequence of real numbers such that <math><mn>1</mn><mo><</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math> and <math><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></munder><mo>⁡</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mo>+</mo><mo>∞</mo></math>. In particular, any dynamical system can be remetrized in order to significantly control the distance between points by their initial distance.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"370 ","pages":"Article 109419"},"PeriodicalIF":0.6000,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topology and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166864125002172","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Main aim of this article is to prove that for any continuous function

f : X \to X

, where X is metrizable (or, more generally, for any family

F

of such functions, with one extra condition), there exists a compatible metric d on X such that the nth iteration of f (more generally, composition of any n functions from

F

) is Lipschitz with constant

a_{n}

where

{(a_{n})}_{n = 1}^{\infty}

is an arbitrarily fixed sequence of real numbers such that

1 < a_{n}

and

\lim_{n \to + \infty} a_{n} = + \infty

. In particular, any dynamical system can be remetrized in order to significantly control the distance between points by their initial distance.

查看原文本刊更多论文

重新计量动力系统以控制时间点的距离

本文的主要目的是证明对于任意连续函数f:X→X，其中X是可度量的（或者更一般地说，对于此类函数的任意族f，有一个附加条件），在X上存在一个相容的度量d，使得f的第n次迭代（更一般地说，由f构成的任意n个函数的复合）是具有常数an的Lipschitz，其中（an）n=1∞是一个任意固定的实数序列，使得1<；an和limn→+∞∑an=+∞。特别地，任何动力系统都可以重新计量，以便通过它们的初始距离显著地控制点之间的距离。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.