Slow entropy type invariants and smooth realization of commuting measure-preserving transformations

Katok, Anatole; Thouvenot, J

doi:10.1016/s0246-0203(97)80094-5

Cited by 87 publications

(112 citation statements)

References 8 publications

Supporting

Mentioning

106

Contrasting

Unclassified

Order By: Relevance

“…It therefore vanishes for many interesting dynamical systems. Following [30,40], we thus also consider the j-th slow volume growth s j (ϕ) = sup It measures the polynomial volume growth of the iterates of the most distorted smooth j-dimensional family of initial data. Note that v j (ϕ), v(ϕ), s j (ϕ), s(ϕ) do not depend on the choice of g and that v dim X (ϕ) = s dim X (ϕ) = 0.…”

Section: Topological Entropy and Volume Growthmentioning

confidence: 99%

Fiberwise volume growth via Lagrangian intersections

Frauenfelder¹,

Schlenk²

2006

Journal of Symplectic Geometry

View full text Add to dashboard Cite

We study certain topological entropy-type growth characteristics of Hamiltonian flows of a special form on the cotangent bundle T * M over a closed Riemannian manifold (M, g). More precisely, the Hamiltonians generating the flows have to be of the form H(t, q, p) = f (|p|) for |p| ≥ 1, and the growth characteristics reflect how the Riemannian volumes of the unit balls in the fibers of T * M grow with the flow. Using a version of Lagrangian Floer homology and recent work of Abbondandolo and Schwarz, we obtain uniform lower bounds for the growth characteristics that depend only on (M, g) or, more precisely, on the homology of the sublevel sets of the energy functional defined by the Riemannian metric g on the space of based loops of Sobolev class W 1,2 in M . These lower bounds, in turn, can be estimated from below by purely topological characteristics of M . Our results in particular refine previous results of Dinaburg, Gromov, Paternain, and Paternain-Petean on the topological entropy of geodesic flows.As an application, we obtain that for Riemannian manifolds, all of whose geodesics are closed (so-called P -manifolds), the fiberwise volume growth of every symplectomorphism in the symplectic isotopy class of the Dehn-Seidel twist is at least linear. This extends the main result of our paper [30] from the class of all currently known P -manifolds to all P -manifolds.

show abstract

Section: Topological Entropy and Volume Growthmentioning

confidence: 99%

Fiberwise volume growth via Lagrangian intersections

Frauenfelder¹,

Schlenk²

2006

Journal of Symplectic Geometry

View full text Add to dashboard Cite

show abstract

“…We introduce the notion of slow metric entropy following the work of Katok and Thouvenot [1]. In this paper they used the slow entropy as a technical tool to obtain smooth realizations of commuting measure preserving transformations.…”

Section: Slow Metric Entropymentioning

confidence: 99%

“…A crucial property of lower and upper slow metric entropies is that they are invariants under an isomorphism of measurable spaces (see [1] for details).…”

Section: Slow Metric Entropymentioning

confidence: 99%

“…Unfortunately, one of the results we stated is not true. In addition, we did not mention an important work in the area by Katok and Thouvenot [1] in which they introduced the notion of slow metric entropy based on an approach that utilizes the Hamming metrics. While our scaled metric entropy may not coincide with the slow metric entropy and they have different properties, the relations between them are instructive and help better understand the rescaling phenomena in dynamics.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Erratum to: Scaled Entropy for Dynamical Systems

Yun

Pesin

2016

J Stat Phys

View full text Add to dashboard Cite

In [4] using the Carathéodory approach as described in [3], we introduced the notions of scaled topological and scaled metric entropies and we described some of their basic properties. Unfortunately, one of the results we stated is not true. In addition, we did not mention an important work in the area by Katok and Thouvenot [1] in which they introduced the notion of slow metric entropy based on an approach that utilizes the Hamming metrics. While our scaled metric entropy may not coincide with the slow metric entropy and they have different properties, the relations between them are instructive and help better understand the rescaling phenomena in dynamics. This note is intended to cover the gap and to correct a mistake in our paper. Scaled Topological EntropyWe begin by briefly recalling the definitions of scaled topological and scaled metric entropies. Consider a continuous transformation T : X → X of a compact metric space X equipped with metric d. A sequence of positive numbers a = {a(n)} n≥1 is a scaled sequence if it is monotonically increasing to infinity.The online version of the original article can be found under

show abstract

“…Most of these examples have the property that for each (k, l), σ k τ l has positive entropy and the directional entropies in every direction including irrational directions are well defined, and moreover they are continuous [5,21,23]. It is not hard to construct examples whose directional entropies are all zero, but their complexities are very different and quite nontrivial [19]. Also there are many physical models showing intermediate chaotic behavior which have recently received a great deal of study [25,28].…”

Section: Introductionmentioning

confidence: 99%

Entropy dimension of topological dynamical systems

Dou

Huang

Park

2011

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We introduce the notion of topological entropy dimension to measure the complexity of entropy zero systems. It measures the superpolynomial, but subexponential, growth rate of orbits. We also introduce the dimension set, D(X, T ) ⊂ [0, 1], of a topological dynamical system to study the complexity of its factors. We construct a minimal example whose dimension set consists of one number. This implies the property that every nontrivial open cover has the same entropy dimension. This notion for zero entropy systems corresponds to the K-mixing property in measurable dynamics and to the uniformly positive entropy in topological dynamics for positive entropy systems. Using the entropy dimension, we are able to discuss the disjointness between the entropy zero systems. Properties of entropy generating sequences and their dimensions have been investigated.

show abstract

Slow entropy type invariants and smooth realization of commuting measure-preserving transformations

Abstract: Slow entropy type invariants and smooth realization of commuting measure-preserving transformations

Cited by 87 publications

References 8 publications

Fiberwise volume growth via Lagrangian intersections

Fiberwise volume growth via Lagrangian intersections

Erratum to: Scaled Entropy for Dynamical Systems

Entropy dimension of topological dynamical systems

Contact Info

Product

Resources

About