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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2409.06661 |
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| _version_ | 1866929494910042112 |
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| author | Hilberdink, Titus |
| author_facet | Hilberdink, Titus |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06661 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classifying Functions via growth rates of repeated iterations Hilberdink, Titus Classical Analysis and ODEs 26A18, 26A12 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. |
| title | Classifying Functions via growth rates of repeated iterations |
| topic | Classical Analysis and ODEs 26A18, 26A12 |
| url | https://arxiv.org/abs/2409.06661 |