Covering number on inhomogeneous graph-directed self-similar sets
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917336422809600 |
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| author | Bárány, Balázs Käenmäki, Antti Nissinen, Petteri |
| author_facet | Bárány, Balázs Käenmäki, Antti Nissinen, Petteri |
| contents | For a strongly connected inhomogeneous graph-directed self-similar set $K^C$ satisfying the strong open set condition, we characterize the asymptotic behaviour of the $r$-covering number $N_r(K^C)$ as $r \downarrow 0$ in terms of the Minkowski dimension $s_0(G)$ of the attractor. If $\int_0^\infty e^{-s_0(G)t}N_{e^{-t}}(C_i)\,\mathrm{d} t<\infty$ for all vertices $i$, then $e^{-s_0(G)t}N_{e^{-t}}(K^C)$ has a limit as $t\to\infty$, which is a positive constant when the log-contraction group $G_M$ is $\mathbb{R}$ and a positive periodic function when $G_M$ is a lattice; if the integral diverges for some $i$, the limit is infinite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16263 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Covering number on inhomogeneous graph-directed self-similar sets Bárány, Balázs Käenmäki, Antti Nissinen, Petteri Dynamical Systems Classical Analysis and ODEs Primary 28A80, Secondary 37C45, 37D35 For a strongly connected inhomogeneous graph-directed self-similar set $K^C$ satisfying the strong open set condition, we characterize the asymptotic behaviour of the $r$-covering number $N_r(K^C)$ as $r \downarrow 0$ in terms of the Minkowski dimension $s_0(G)$ of the attractor. If $\int_0^\infty e^{-s_0(G)t}N_{e^{-t}}(C_i)\,\mathrm{d} t<\infty$ for all vertices $i$, then $e^{-s_0(G)t}N_{e^{-t}}(K^C)$ has a limit as $t\to\infty$, which is a positive constant when the log-contraction group $G_M$ is $\mathbb{R}$ and a positive periodic function when $G_M$ is a lattice; if the integral diverges for some $i$, the limit is infinite. |
| title | Covering number on inhomogeneous graph-directed self-similar sets |
| topic | Dynamical Systems Classical Analysis and ODEs Primary 28A80, Secondary 37C45, 37D35 |
| url | https://arxiv.org/abs/2307.16263 |