Classifying Functions via growth rates of repeated iterations

TL;DR

This paper introduces a novel classification system for real functions based on the growth rates of their repeated iterations and their inverses, revealing large gaps and potential continuum of classes.

Contribution

It develops a new framework for classifying functions by their iteration growth rates and proves properties including a uniqueness aspect of these classes.

Findings

Functions can be grouped into classes based on iteration growth rates.

There are large gaps between these classes, possibly forming a continuum.

The classification includes functions like linear, exponential, and iterated exponential.

Abstract

In this paper we develop a classification of real functions based on growth rates of repeated iteration. We show how functions are naturally distinguishable when considering inverses of repeated iterations. For example, $n+2\to 2n\to 2^n\to 2^{\cdot^{\cdot^2}}$ ($n$-times) etc. and their inverse functions $x-2, x/2, \log x/\log 2,$ etc. Based on this idea and some regularity conditions we define classes of functions, with $x+2$, $2x$, $2^x$ in the first three classes. We prove various properties of these classes which reveal their nature, including a `uniqueness' property. We exhibit examples of functions lying between consecutive classes and indicate how this implies these gaps are very `large'. Indeed, we suspect the existence of a continuum of such classes.