Period Lengths for Iterated Functions

TL;DR

This paper analyzes the expected period lengths of iterated functions under random maps, showing that the expected order is asymptotically smaller than the expected product of cycle lengths.

Contribution

It provides an asymptotic approximation for the expected order in random maps, highlighting a significant difference from the expected product of cycle lengths.

Findings

Expected order is asymptotically smaller than product of cycle lengths

Provides asymptotic approximation for expected order in random maps

Highlights difference between order and cycle length product expectations

Abstract

For random maps, the expected value of the order (i.e. the period of the sequence of compositional iterates) is approximated asymptotically. It is much smaller than the expected value for the product of the cycle lengths.