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Masaki Tsukamoto

Publications and source records attributed to Masaki Tsukamoto.

At least 19 recordsLinked to original sources
arXiv2026

Comparison of two approaches to weighted topological entropy

We compare the Feng--Huang and covering definitions of weighted topological entropy for equivariant continuous maps between dynamical systems. The two quantities agree on the whole space by their variational principles, but need not agree on non-invariant subsets. We prove that the Feng--Huang entropy is bounded above by the covering entropy on every subset, without assuming invariance. The proof does not rely on measure theory. Our main result provides a new perspective on the two relative weighted variational principles.

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arXiv2026

Weighted topological entropy and intersecting random translates of Bedford--McMullen carpets

We establish a relativised variational principle for the Feng--Huang weighted topological entropy associated with a factor map between dynamical systems. Combined with a recent theorem of Yin, this yields an almost-everywhere equivalence between the Feng--Huang entropy and its combinatorial version on fibers. As an application, we compute the Hausdorff dimension of the intersection of random translates of two Bedford--McMullen carpets. The resulting formula extends the Kenyon--Peres formula from the self-similar to the self-affine setting, and also points to a new problem concerning random matrix products.

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arXiv2025

A Lipschitz Refinement of the Multidimensional Bebutov--Kakutani Dynamical Embedding Theorem

We prove that a continuous action of $\mathbb{R}^n$ on a compact metrizable space equivariantly embeds into the shift action on the space of one-Lipschitz functions from $\mathbb{R}^n$ to $[0,1]$ if and only if the set of fixed points topologically embeds in $[0,1]$. This is a Lipschitz refinement of classical dynamical embedding theorems of Bebutov, Kakutani, Jaworski and Chen.

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arXiv2025

Rate distortion dimension and ergodic decomposition for $\mathbb{R}^d$-actions

Rate distortion dimension describes the theoretical limit of lossy data compression methods as the distortion bound goes to zero. It was originally introduced in the context of information theory, and recently it was discovered that it has an intimate connection to Gromov's theory of mean dimension of dynamical systems. This paper studies the behavior of rate distortion dimension of $\mathbb{R}^d$-actions under ergodic decomposition. Our main theorems provide natural convexity and concavity of upper and lower rate distortion dimensions under convex combination of invariant probability measures. We also present examples which clarify the validity and limitations of the theorems.

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arXiv2024

Rate distortion dimension of random Brody curves

The main purpose of this paper is to propose an ergodic theoretic approach to the study of entire holomorphic curves. Brody curves are one-Lipschitz holomorphic maps from the complex plane to the complex projective space. They naturally form a dynamical system, and "random Brody curves" in the title refers to invariant probability measures on it. We study their geometric and dynamical properties. Given an invariant probability measure $μ$ on the space of Brody curves, our first main theorem claims that its rate distortion dimension is bounded by the integral of a "potential function" over $μ$. This result is analogous to the Ruelle inequality of smooth ergodic theory. Our second main theorem claims that there exists a rich variety of invariant probability measures attaining equality in this "Ruelle inequality for Brody curves". The main tools of the proofs are the deformation theory of Brody curves and the variational principle for mean dimension with potential. This approach is motivated by the theory of thermodynamic formalism for Axiom A diffeomorphisms.

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arXiv2023

Application of waist inequality to entropy and mean dimension: II

Let $X$ be a full-shift on the alphabet $[0, 1]^a$ and let $(Y, S)$ be an arbitrary dynamical system. We prove that any equivariant continuous map from $X$ to $Y$ has conditional metric mean dimension not less than $a-\mathrm{mdim}(Y, S)$. This solves a problem posed in a paper of Shi--Tsukamoto (2023).

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arXiv2023

Variational principle for mean dimension with potential of $\mathbb{R}^d$-actions: I

We develop a variational principle for mean dimension with potential of $\mathbb{R}^d$-actions. We prove that mean dimension with potential is bounded from above by the supremum of the sum of rate distortion dimension and a potential term. A basic strategy of the proof is the same as the case of $\mathbb{Z}$-actions. However measure theoretic details are more involved because $\mathbb{R}^d$ is a continuous group. We also establish several basic properties of metric mean dimension with potential and mean Hausdorff dimension with potential for $\mathbb{R}^d$-actions.

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arXiv2022

Application of waist inequality to entropy and mean dimension

Waist inequality is a fundamental inequality in geometry and topology. We apply it to the study of entropy and mean dimension of dynamical systems. We consider equivariant continuous maps between dynamical systems and assume that the mean dimension of the domain is larger than the mean dimension of the target. We exhibit several situations for which the maps necessarily have positive conditional metric mean dimension. This study has interesting consequences to the theory of topological conditional entropy. In particular it sheds new light on a celebrated result of Lindenstrauss and Weiss about minimal dynamical systems non-embeddable in the shift on the Hilbert cube.

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arXiv2022

Mean Hausdorff dimension of some infinite dimensional fractals

Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff dimension of homogeneous sets in the circle. (2) The coincidence of Hausdorff and Minkowski dimensions for self-similar sets. (3) The calculation of Hausdorff dimension of Bedford--McMullen carpets. We develop their analogues for mean Hausdorff dimension: (1) The calculation of mean Hausdorff dimension of homogeneous systems in the infinite dimensional torus. (2) The coincidence of mean Hausdorff dimension and metric mean dimension for self-similar systems. (3) The calculation of mean Hausdorff dimension of infinite dimensional carpets.

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arXiv2022

On an analogue of the Hurewicz theorem for mean dimension

The Hurewicz theorem is a fundamental result in classical dimension theory concerning continuous maps which lower topological dimension. We study whether or not its analogue holds for mean dimension of dynamical systems. Our first main result shows that an analogue of the Hurewicz theorem does not hold for mean dimension in general. Our second main result shows that it holds true if a base system has zero mean dimension.

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arXiv2021

New approach to weighted topological entropy and pressure

Motivated by fractal geometry of self-affine carpets and sponges, Feng--Huang (2016) introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of Feng--Huang. The equivalence of the two definitions seems highly nontrivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford--McMullen carpet in purely topological terms.

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arXiv2021

Divergent coindex sequence for dynamical systems

When a finite group freely acts on a topological space, we can define its index and coindex. They roughly measure the size of the given action. We explore the interaction between this index theory and topological dynamics. Given a fixed-point free dynamical system, the set of $p$-periodic points admits a natural free action of $\mathbb{Z}/p\mathbb{Z}$ for each prime number $p$. We are interested in the growth of its index and coindex as $p\to \infty$. Our main result shows that there exists a fixed-point free dynamical system having the divergent coindex sequence. This solves a problem posed by [TTY20].

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arXiv2021

Remark on the local nature of metric mean dimension

Metric mean dimension is a metric invariant of dynamical systems. It is a dynamical analogue of Minkowski dimension of metric spaces. We explain that old ideas of Bowen (1972) can be used for clarifying the local nature of metric mean dimension. We also explain the generalization to $\mathbb{R}^D$-actions and an illustrating example.

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arXiv2020

$G$-index, topological dynamics and marker property

Given an action of a finite group $G$, we can define its index. The $G$-index roughly measures a size of the given $G$-space. We explore connections between the $G$-index theory and topological dynamics. For a fixed-point free dynamical system, we study the $\mathbb{Z}_p$-index of the set of $p$-periodic points. We find that its growth is at most linear in $p$. As an application, we construct a free dynamical system which does not have the marker property. This solves a problem which has been open for several years.

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arXiv2019

Symbolic dynamics in mean dimension theory

Furstenberg (1967) calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to $\mathbb{Z}^2$-subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and mean Hausdorff dimension of $\mathbb{Z}^2$-subshifts with respect to a subaction of $\mathbb{Z}$. The resulting formula is quite analogous to Furstenberg's theorem. We also calculate the rate distortion dimension of $\mathbb{Z}^2$-subshifts in terms of Kolmogorov-Sinai entropy.

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arXiv2019

Double variational principle for mean dimension

We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

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arXiv2019

Double variational principle for mean dimension with potential

This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of topological pressure. We consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function. We prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property. The basic idea of the proof is a dynamicalization of geometric measure theory.

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