Classifying Functions via growth rates of repeated iterations
| Autor: | Hilberdink, Titus |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 23 pages. To appear in Fundamenta Mathematicae |
| Databáze: | arXiv |
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