Lectures on Foliation Dynamics

Hurder, Steven

doi:10.1007/978-3-0348-0871-2_3

Cited by 8 publications

(12 citation statements)

References 163 publications

(266 reference statements)

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“…In 1988, Walczak et al [ 74 ] introduced the geometric entropy (or foliation entropy) to study a foliation dynamics, which can be considered as a generalization of TopEn of a single group [ 75 , 76 , 77 ]. Recently, Rong and Shang [ 10 ] proposed geometric entropy for time-series based on the multiscale method (see Section 2.10 ) and the original definition of geometric entropy provided by Walczak.…”

Section: Building the Universe Of Entropiesmentioning

confidence: 99%

The Entropy Universe

Ribeiro

Henriques

Castro

et al. 2021

Entropy

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About 160 years ago, the concept of entropy was introduced in thermodynamics by Rudolf Clausius. Since then, it has been continually extended, interpreted, and applied by researchers in many scientific fields, such as general physics, information theory, chaos theory, data mining, and mathematical linguistics. This paper presents The Entropy Universe, which aims to review the many variants of entropies applied to time-series. The purpose is to answer research questions such as: How did each entropy emerge? What is the mathematical definition of each variant of entropy? How are entropies related to each other? What are the most applied scientific fields for each entropy? We describe in-depth the relationship between the most applied entropies in time-series for different scientific fields, establishing bases for researchers to properly choose the variant of entropy most suitable for their data. The number of citations over the past sixteen years of each paper proposing a new entropy was also accessed. The Shannon/differential, the Tsallis, the sample, the permutation, and the approximate entropies were the most cited ones. Based on the ten research areas with the most significant number of records obtained in the Web of Science and Scopus, the areas in which the entropies are more applied are computer science, physics, mathematics, and engineering. The universe of entropies is growing each day, either due to the introducing new variants either due to novel applications. Knowing each entropy’s strengths and of limitations is essential to ensure the proper improvement of this research field.

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Section: Building the Universe Of Entropiesmentioning

confidence: 99%

The Entropy Universe

Ribeiro

Henriques

Castro

et al. 2021

Entropy

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“…We also have the subclass of nonexponential growth type, where M 0 has quasi-polynomial growth type if there exists d ≥ 0 such that Gr(M 0 , s) s d . The growth type of a leaf of a foliation or lamination is an entropy-type invariant of its dynamics, as discussed in [21].…”

Section: Growth Slow Entropy and Hausdorff Dimensionmentioning

confidence: 99%

Perspectives on Kuperberg flows

Hurder¹,

Rechtman²

2016

Preprint

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The "Seifert Conjecture" asks, "Does every non-singular vector field on the 3-sphere S 3 have a periodic orbit?" In a celebrated work, Krystyna Kuperberg gave a construction of a smooth aperiodic vector field on a plug, which is then used to construct counter-examples to the Seifert Conjecture for smooth flows on the 3-sphere, and on compact 3-manifolds in general. The dynamics of the flows in these plugs have been extensively studied, with more precise results known in special "generic" cases of the construction. Moreover, the dynamical properties of smooth perturbations of Kuperberg's construction have been considered. In this work, we recall some of the results obtained to date for the Kuperberg flows and their perturbations. Then the main point of this work is to focus attention on how the known results for Kuperberg flows depend on the assumptions imposed on the flows, and to discuss some of the many interesting questions and problems that remain open about their dynamical and ergodic properties. Contents

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“…Recall from definition 2.9 (see also [16]) that a pseudogroup Γ on a Polish space Ω isexpansive if for all w = w ∈ Ω with d(w, w ) < there exists γ ∈ Γ with w, w ∈ dom h such that d(γ(w), γ(w )) ≥ , where d is a metric on Ω. We now prove Theorem 1.2, which shows that if Ω is compact, and Γ is compactly generated and -expansive, then the pseudogroup dynamical system (Ω, Γ) can be equivariantly embedded into the space of pointed trees X n , for n large enough.…”

Section: Universal Space For Compactly Generated Pseudogroupsmentioning

confidence: 99%

Suspensions of Bernoulli shifts

Rojo

Lukina

2013

Dynamical Systems

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We show that for a given finitely generated group, its Bernoulli shift space can be equivariantly embedded as a subset of a space of pointed trees with Gromov-Hausdorff metric and natural partial action of a free group. Since the latter can be realised as a transverse space of a foliated space with leaves Riemannian manifolds, this embedding allows us to obtain a suspension of such Bernoulli shift. By a similar argument we show that the space of pointed trees is universal for compactly generated expansive pseudogroups of transformations.

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Lectures on Foliation Dynamics

Cited by 8 publications

References 163 publications

The Entropy Universe

The Entropy Universe

Perspectives on Kuperberg flows

Suspensions of Bernoulli shifts

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