Dynamical metric order

TL;DR

The paper introduces the dynamical metric order, a new complexity measure for continuous maps on compact metric spaces, especially useful for spaces with infinite box-counting dimension.

Contribution

It defines and studies the properties of dynamical metric order, linking it to metric mean dimension and providing new insights into the complexity of certain dynamical systems.

Findings

Calculates dynamical metric order for various classes of maps.

Shows the concept refines complexity estimates for maps with infinite box-counting dimension.

Establishes a variational principle with guaranteed equilibrium states.

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.