How many times can a function be iterated?

2009/01/21 by Massimo Gobbino, Gobbino, Massimo, Robert Samuel Simon +1
Mathematics · #37B99 #54H20 #55N05 #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #math.AT #math.DS #msc:37B99 #msc:54H20 #msc:55N05

paper · pdf · doi:10.48550/arxiv.0901.3230

28 pages, 8 figures, section with applications to game theory rewritten, references updated. To appear on the Journal of Difference Equations and Applications

openalex publication_date 2009/01/21 · arxiv created 2011/11/07 · arxiv updated 2011/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

Let C be a closed subset of a topological space X, and let f : C --> X. Let us assume that f is continuous and f(x) lies in C for every x in the boundary of C. How many times can one iterate f? This paper provides estimates on the number of iterations and examples of their optimality. In particular we show how some topological properties of f, C, X are related to the maximal number of iterations, both in the case of functions and in the more general case of set-valued maps.

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