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arXiv · 2603.28401

Dynamical metric order

Maria Carvalho·Fagner B. Rodrigues
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Abstract

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.

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Maria Carvalho, Fagner B. Rodrigues. 2026-03-30. Dynamical metric order. https://arxiv.org/abs/2603.28401

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