Scale Recurrence Lemma and Dimension Formula for Cantor Sets in the Complex Plane
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866916487193690112 |
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| author | Moreira, Carlos Gustavo T. de A. Zamudio, Alex Mauricio |
| author_facet | Moreira, Carlos Gustavo T. de A. Zamudio, Alex Mauricio |
| contents | We will prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz \cite{MY} for Cantor sets in the complex plane. We then use this new recurrence lemma, together with the ideas in \cite{M}, to prove that under the right hypothesis for the Cantor sets $K_1,...,K_n$ and the function $h:\mathbb{C}^{n}\to \mathbb{R}^{l}$, the following formula holds \[HD(h(K_1\times K_2 \times ...\times K_n))=\min \{l,HD(K_1)+...+HD(K_n)\}.\] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1911_03041 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Scale Recurrence Lemma and Dimension Formula for Cantor Sets in the Complex Plane Moreira, Carlos Gustavo T. de A. Zamudio, Alex Mauricio Dynamical Systems We will prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz \cite{MY} for Cantor sets in the complex plane. We then use this new recurrence lemma, together with the ideas in \cite{M}, to prove that under the right hypothesis for the Cantor sets $K_1,...,K_n$ and the function $h:\mathbb{C}^{n}\to \mathbb{R}^{l}$, the following formula holds \[HD(h(K_1\times K_2 \times ...\times K_n))=\min \{l,HD(K_1)+...+HD(K_n)\}.\] |
| title | Scale Recurrence Lemma and Dimension Formula for Cantor Sets in the Complex Plane |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/1911.03041 |