Inhomogeneous attractors and box dimension
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914866490507264 |
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| author | Fraser, Jonathan M. |
| author_facet | Fraser, Jonathan M. |
| contents | Iterated function systems (IFSs) are one of the most important tools for building examples of fractal sets exhibiting some kind of `approximate self-similarity'. Examples include self-similar sets, self-affine sets etc. A beautiful variant on the standard IFS model was introduced by Barnsley and Demko in 1985 where one builds an \emph{inhomogeneous} attractor by taking the closure of the orbit of a fixed compact condensation set under a given standard IFS. In this expository article I will discuss the dimension theory of inhomogeneous attractors, giving several examples and some open questions. I will focus on the upper box dimension with emphasis on how to derive good estimates, and when these estimates fail to be sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08404 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inhomogeneous attractors and box dimension Fraser, Jonathan M. Dynamical Systems Metric Geometry 28A80, 26A18, 37F32 Iterated function systems (IFSs) are one of the most important tools for building examples of fractal sets exhibiting some kind of `approximate self-similarity'. Examples include self-similar sets, self-affine sets etc. A beautiful variant on the standard IFS model was introduced by Barnsley and Demko in 1985 where one builds an \emph{inhomogeneous} attractor by taking the closure of the orbit of a fixed compact condensation set under a given standard IFS. In this expository article I will discuss the dimension theory of inhomogeneous attractors, giving several examples and some open questions. I will focus on the upper box dimension with emphasis on how to derive good estimates, and when these estimates fail to be sharp. |
| title | Inhomogeneous attractors and box dimension |
| topic | Dynamical Systems Metric Geometry 28A80, 26A18, 37F32 |
| url | https://arxiv.org/abs/2407.08404 |