Mean topological dimension

Lindenstrauss, Elon; Weiss, Benjamin

doi:10.1007/bf02810577

Cited by 328 publications

(425 citation statements)

References 9 publications

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“…Again the limit exists by (a simpler version of) [LW,Theorem 6.1] and it is independent of the choice of (F n ). Set h µ = sup P h(P).…”

Section: They Imply the Following Standard Resultmentioning

confidence: 99%

Fuglede–Kadison determinants and entropy for actions of discrete amenable groups

Deninger¹

2006

J. Amer. Math. Soc.

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In 1990, Lind, Schmidt, and Ward gave a formula for the entropy of certain Z n \mathbb {Z}^n -dynamical systems attached to Laurent polynomials P P , in terms of the (logarithmic) Mahler measure of P P . We extend the expansive case of their result to the noncommutative setting where Z n \mathbb {Z}^n gets replaced by suitable discrete amenable groups. Generalizing the Mahler measure, Fuglede–Kadison determinants from the theory of group von Neumann algebras appear in the entropy formula.

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“…Again the limit exists by (a simpler version of) [LW,Theorem 6.1] and it is independent of the choice of (F n ). Set h µ = sup P h(P).…”

Section: They Imply the Following Standard Resultmentioning

confidence: 99%

Fuglede–Kadison determinants and entropy for actions of discrete amenable groups

Deninger¹

2006

J. Amer. Math. Soc.

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“…This idea leads to a version of 'slow dimension growth' which in turn is closely related to the notion of mean dimension introduced by Lindenstrauss and Weiss in [18].…”

Section: Introductionmentioning

confidence: 99%

Rokhlin Dimension and C*-Dynamics

Hirshberg

Winter

Zacharias

2015

Commun. Math. Phys.

159

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Abstract:We develop the concept of Rokhlin dimension for integer and for finite group actions on C * -algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Z-stable C * -algebras, where Z denotes the Jiang-Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve Z-stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic Z d -actions on compact metrizable and finite dimensional spaces.

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“…The significance of mean dimension in topological dynamics was clarified by the works of Lindenstrauss and Weiss [11,10]. Here is a sample of their results [10,Theorem 5.1].…”

Section: Mean Dimension Formulamentioning

confidence: 97%

“…So the mean dimension is a topological invariant. Next we introduce metric mean dimension (Lindenstrauss-Weiss [11,Section 4]). For any ε > 0 the function h(Ω) := log #(X, d Ω , ε) also satisfies the conditions of the Ornstein-Weiss lemma.…”

Section: Mean Dimensionmentioning

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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Mean topological dimension

Cited by 328 publications

References 9 publications

Fuglede–Kadison determinants and entropy for actions of discrete amenable groups

Fuglede–Kadison determinants and entropy for actions of discrete amenable groups

Rokhlin Dimension and C*-Dynamics

Mean dimension of the dynamical system of Brody curves

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