Iteration entropy

TL;DR

This paper explores the concept of entropy in iterated functions on finite sets, revealing asymptotic behavior related to the set size and cycle structure, with implications for understanding randomness in such systems.

Contribution

It introduces an analysis of iteration entropy for functions on finite sets, extending known results from permutations to general functions.

Findings

Entropy approaches log_2(n) minus cycle length entropy for permutations

Similar asymptotic approximation holds for general functions

Provides insights into the randomness of iterated functions

Abstract

We apply a common measure of randomness, the entropy, in the context of iterated functions on a finite set with n elements. For a permutation, it turns out that this entropy is asymptotically (for a growing number of iterations) close to \log_2(n) minus the entropy of the vector of its cycle lengths. For general functions, a similar approximation holds.