Dynamical metric order
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910120224489472 |
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| author | Carvalho, Maria Rodrigues, Fagner B. |
| author_facet | Carvalho, Maria Rodrigues, Fagner B. |
| contents | We introduce the notion of dynamical metric order of a continuous map on a compact metric space, study its basic properties, and compute it for several classes of maps. This concept which is a counterpart of the metric mean dimension with the role of the box-counting dimension being played by the metric order. It is devised for maps acting on spaces with infinite box-counting dimension but finite metric order. For example, it brings forward new information about full shifts whose alphabets have infinite box-counting dimension; and provides a sharper estimate of complexity for the induced map determined by a continuous transformation on a compact metric space, whose upper metric mean dimension is known to admit only two values (zero or infinity). We also show that it satisfies a variational principle where maximization is taken over the space of invariant probability measures and whose equilibrium states always exist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28401 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dynamical metric order Carvalho, Maria Rodrigues, Fagner B. Dynamical Systems 28D20, 37A05, 37C45, 37B40, 37D35 We introduce the notion of dynamical metric order of a continuous map on a compact metric space, study its basic properties, and compute it for several classes of maps. This concept which is a counterpart of the metric mean dimension with the role of the box-counting dimension being played by the metric order. It is devised for maps acting on spaces with infinite box-counting dimension but finite metric order. For example, it brings forward new information about full shifts whose alphabets have infinite box-counting dimension; and provides a sharper estimate of complexity for the induced map determined by a continuous transformation on a compact metric space, whose upper metric mean dimension is known to admit only two values (zero or infinity). We also show that it satisfies a variational principle where maximization is taken over the space of invariant probability measures and whose equilibrium states always exist. |
| title | Dynamical metric order |
| topic | Dynamical Systems 28D20, 37A05, 37C45, 37B40, 37D35 |
| url | https://arxiv.org/abs/2603.28401 |