On exponential separation of analytic self-conformal sets on the real line
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| Format: | Preprint |
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2025
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| _version_ | 1866915892013563904 |
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| author | Bárány, Balázs Kolossváry, István Troscheit, Sascha |
| author_facet | Bárány, Balázs Kolossváry, István Troscheit, Sascha |
| contents | In a recent article, Rapaport showed that there is no dimension drop for exponentially separated analytic IFSs on the real line. We show that the set of such exponentially separated IFSs in the space of analytic IFSs contains an open and dense set in the $\mathcal{C}^2$ topology. Moreover, we give a sufficient condition for the IFS to be exponentially separated which allows us to construct explicit examples which are exponentially separated. The key technical tool is the introduction of the \emph{dual IFS} which we believe has significant interest in its own right. As an application we also characterise when an analytic IFS can be conjugated to a self-similar IFS. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_07888 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On exponential separation of analytic self-conformal sets on the real line Bárány, Balázs Kolossváry, István Troscheit, Sascha Dynamical Systems Classical Analysis and ODEs Metric Geometry Primary: 28A80, Secondary: 37C45 In a recent article, Rapaport showed that there is no dimension drop for exponentially separated analytic IFSs on the real line. We show that the set of such exponentially separated IFSs in the space of analytic IFSs contains an open and dense set in the $\mathcal{C}^2$ topology. Moreover, we give a sufficient condition for the IFS to be exponentially separated which allows us to construct explicit examples which are exponentially separated. The key technical tool is the introduction of the \emph{dual IFS} which we believe has significant interest in its own right. As an application we also characterise when an analytic IFS can be conjugated to a self-similar IFS. |
| title | On exponential separation of analytic self-conformal sets on the real line |
| topic | Dynamical Systems Classical Analysis and ODEs Metric Geometry Primary: 28A80, Secondary: 37C45 |
| url | https://arxiv.org/abs/2509.07888 |