Large dynamics of Yang–Mills theory: mean dimension formula

Tsukamoto, Masaki

doi:10.1007/s11854-018-0014-2

Cited by 15 publications

(11 citation statements)

References 35 publications

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“…Metric mean dimension was introduced in [23] and proved to be, when calculated with respect to any compatible metric, an upper bound for the topological mean dimension. It was recently successfully used in [52] and [53] to obtain formulas for mean dimension of dynamical systems arising from geometric analysis.…”

Section: Discussionmentioning

confidence: 99%

Metric mean dimension and analog compression

Gutman¹,

Śpiewak²

2020

Preprint

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<div>Wu and Verdú developed a theory of almost lossless analog compression, where one imposes various regularity conditions on the compressor and the decompressor with the input signal being modelled by a (typically infinite-entropy) stationary stochastic process. In this work we consider all stationary stochastic processes with trajectories in a prescribed set of (bi-)infinite sequences and find uniform lower and upper bounds for certain compression rates in terms of metric mean dimension and mean box dimension. An essential tool is the recent Lindenstrauss-Tsukamoto variational principle expressing metric mean dimension in terms of rate-distortion functions. We obtain also lower bounds on compression rates for a fixed stationary process in terms of the rate-distortion dimension rates and study several examples.</div>

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Section: Discussionmentioning

confidence: 99%

Metric mean dimension and analog compression

Gutman¹,

Śpiewak²

2020

Preprint

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“…Although our investigation in this direction has just started, the result in §6 seems to suggest a high potential of this research direction. It is desirable to study geometric examples in [Gro99,T18a,T18b] from the viewpoint of the double variational principle.…”

Section: About Stepmentioning

confidence: 99%

Double variational principle for mean dimension

Lindenstrauss

Tsukamoto

2019

Geom. Funct. Anal.

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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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“…It is hopeless to prove the optimal bound (1.5) by this method, and there also exist several problems facing similar difficulties. For example, Gromov [ In order to overcome these difficult situations, we started to develop a completely new approach in [18]. The paper [18] examined a new technique in the context of Yang-Mills gauge theory.…”

Section: Theorem 12 (Main Theorem)mentioning

confidence: 99%

“…For example, Gromov [ In order to overcome these difficult situations, we started to develop a completely new approach in [18]. The paper [18] examined a new technique in the context of Yang-Mills gauge theory. Based on this experience, now we attack to Brody curves.…”

Section: Theorem 12 (Main Theorem)mentioning

confidence: 99%

Mean dimension of the dynamical system of Brody curves

Tsukamoto

2017

Invent. math.

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Abstract. Mean dimension measures the size of an infinite dimensional dynamical system. Brody curves are one-Lipschitz entire holomorphic curves in the projective space, and they form a topological dynamical system. Gromov started the problem of estimating its mean dimension in the paper of 1999. We solve this problem. Namely we prove the exact mean dimension formula of the dynamical system of Brody curves. Our formula expresses the mean dimension by the energy density of Brody curves. The proof is based on a novel application of the metric mean dimension theory of Lindenstrauss and Weiss. Mean dimension formulaThis is the local Lipschitz constant. For a unit tangent vector u ∈ T z C the Fubini-Study length of df (u) ∈ T f (z) CP N is equal to the spherical derivative |df |(z) Recently new waves have begun, and Brody curves attract new interests of several authors [1,3,4,5,6,12,14,15,19]. One source of new interests is the work of Gromov [8]. He introduced a new topological invariant of dynamical systems called mean dimension, and proposed a program to study many infinite dimensional dynamical systems in geometric analysis from the viewpoint of this invariant. The dynamical system consisting of Brody curves is the simplest example in this program. The purpose of our paper is.

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Large dynamics of Yang–Mills theory: mean dimension formula

Cited by 15 publications

References 35 publications

Metric mean dimension and analog compression

Metric mean dimension and analog compression

Double variational principle for mean dimension

Mean dimension of the dynamical system of Brody curves

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