A note on the Banach limit-based proof of Elton's ergodic theorem for iterated function systems

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-07-08 DOI:10.1016/j.topol.2025.109519

R. Medhi, P. Viswanathan

引用次数: 0

Abstract

Elton's ergodic theorem for a contractive Iterated Function System (IFS) is a foundational result with wide-ranging applications. The brief article (Forte and Mendivil (1998) [5]) presents a short and elegant proof of this theorem using Banach limit techniques. In this note, we identify a subtle but significant flaw in that proof, specifically, an unjustified inference from the uniqueness of Banach limits to actual convergence. To address this, we provide a corrected and rigorous proof using the connection between almost convergence and Cesàro convergence. This corrected approach may also provide a template for establishing ergodic theorems for IFS beyond the classical contractive setting.

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关于迭代函数系统Elton遍历定理的Banach极限证明的注解

压缩迭代函数系统（IFS）的Elton遍历定理是一个具有广泛应用的基础结果。这篇简短的文章（Forte和Mendivil(1998)[5]）用巴拿赫极限技术给出了这个定理的一个简短而优雅的证明。在本文中，我们在该证明中发现了一个微妙但重要的缺陷，特别是从Banach极限的唯一性到实际收敛的不合理推论。为了解决这个问题，我们使用近似收敛和Cesàro收敛之间的联系提供了一个正确的和严格的证明。这种修正的方法也可以为建立超越经典收缩设定的IFS遍历定理提供模板。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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.