How many times can a function be iterated?

TL;DR

This paper investigates the maximum number of iterations possible for continuous functions and set-valued maps on closed subsets of topological spaces, providing estimates and examples of optimality based on topological properties.

Contribution

It introduces bounds on iteration counts for functions and set-valued maps, linking these bounds to topological characteristics of the space and the maps.

Findings

Derived estimates for iteration limits

Provided examples demonstrating optimality of bounds

Connected topological properties to iteration counts

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.