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Entropic Properties of Dynamical Systems

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A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (30 October 2016) | Viewed by 30314

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Dr. Tomasz Downarowicz

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Faculty of Mathematics, Wrocław University of Science and Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland
Interests: measure-theoretic entropy; topological entropy; amenable entropy; operator entropy; symbolic extensions; law of series; chaos

Special Issue Information

Dear Colleagues,

Dynamical systems, regardless of the wide range of objects that can be understood under this name, can be classified into three main classes: zero entropy, finite positive entropy, and infinite entropy systems. Zero entropy systems are those with slow (subexponential) complexity (whatever that means), finite positive entropy systems usually reveal some kind of randomness or chaos, and so do infinite entropy systems, but surprisingly they share some abilities of zero entropy systems, for example they can be self-similar (isomorphic to their iterate powers). In recent years, many interesting connections between the above classifications and other dynamical properties have been discovered, especially, when one focuses on more specific families of dynamical systems (subshifts, interval maps, smooth maps on manifolds, actions of particular groups, special flows, non-autonomous systems of certain kinds, etc.). As an example, the relation between entropy and distribution of periodic points has been a frequent subject of study. Additionally, many interesting entropy-like parameters have been invented to study low complexity systems, such as slow entropy, entropy dimension, sequence entropy, etc. Mean dimension, on the other hand, enables one to classify systems with infinite entropy. Some famous conjectures also relate to the above classification, for instance Sarnak’s conjecture or, after Rudolph’s result, Furstenberg’s ×2×3 conjecture (and its generalizations) as well.

In the Special Issue we would like to gather new results in the above spirit: new consequences of positivity or non-positivity of entropy in specific types of dynamical systems, and also the opposite: examples of surprising and counterintuitive coexistence of entropic and non-entropic properties, new tools allowing to classify zero or infinite entropy systems with consequences of such a classification.

Dr. Tomasz Downarowicz
Guest Editor

Manuscript Submission Information

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Keywords

Zero entropy systems: slow entropy, entropy dimension, sequence entropy, Sarnak’s conjecture, distribution of periodic points, self-similarity
Positive entropy systems: consequences of positive entropy, chaos, distributional chaos, law of series, distribution of periodic points
Infinite entropy: mean dimension, distribution of periodic points, self-similarity

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Published Papers (7 papers)

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300 KiB

Open AccessArticle

Variational Principle for Relative Tail Pressure

by Xianfeng Ma and Ercai Chen

Entropy 2017, 19(3), 120; https://doi.org/10.3390/e19030120 - 15 Mar 2017

Cited by 2 | Viewed by 3534

Abstract

We introduce the relative tail pressure to establish a variational principle for continuous bundle random dynamical systems. We also show that the relative tail pressure is conserved by the principal extension. Full article

(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)

284 KiB

Open AccessArticle

Topological Entropy Dimension and Directional Entropy Dimension for ℤ²-Subshifts

by Uijin Jung, Jungseob Lee and Kyewon Koh Park

Entropy 2017, 19(2), 46; https://doi.org/10.3390/e19020046 - 24 Jan 2017

Cited by 4 | Viewed by 4485

Abstract

The notion of topological entropy dimension for a

Z

-action has been introduced to measure the subexponential complexity of zero entropy systems. Given a

Z^{2}

-action, along with a

Z^{2}

-entropy dimension, we also consider a finer notion of directional entropy [...] Read more.

The notion of topological entropy dimension for a

Z

-action has been introduced to measure the subexponential complexity of zero entropy systems. Given a

Z^{2}

-action, along with a

Z^{2}

-entropy dimension, we also consider a finer notion of directional entropy dimension arising from its subactions. The entropy dimension of a

Z^{2}

-action and the directional entropy dimensions of its subactions satisfy certain inequalities. We present several constructions of strictly ergodic

Z^{2}

-subshifts of positive entropy dimension with diverse properties of their subgroup actions. In particular, we show that there is a

Z^{2}

-subshift of full dimension in which every direction has entropy 0. Full article

(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)

274 KiB

Open AccessArticle

Fiber-Mixing Codes between Shifts of Finite Type and Factors of Gibbs Measures

by Uijin Jung

Entropy 2016, 18(12), 428; https://doi.org/10.3390/e18120428 - 30 Nov 2016

Cited by 2 | Viewed by 3781

Abstract

A sliding block code

π : X \to Y

between shift spaces is called fiber-mixing if, for every x and

x^{'}

in X with

y = π (x) = π (x^{'})

, there is [...] Read more.

A sliding block code

π : X \to Y

between shift spaces is called fiber-mixing if, for every x and

x^{'}

in X with

y = π (x) = π (x^{'})

, there is

z \in π^{- 1} (y)

which is left asymptotic to x and right asymptotic to

x^{'}

. A fiber-mixing factor code from a shift of finite type is a code of class degree 1 for which each point of Y has exactly one transition class. Given an infinite-to-one factor code between mixing shifts of finite type (of unequal entropies), we show that there is also a fiber-mixing factor code between them. This result may be regarded as an infinite-to-one (unequal entropies) analogue of Ashley’s Replacement Theorem, which states that the existence of an equal entropy factor code between mixing shifts of finite type guarantees the existence of a degree 1 factor code between them. Properties of fiber-mixing codes and applications to factors of Gibbs measures are presented. Full article

(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)

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282 KiB

Open AccessArticle

Positive Sofic Entropy Implies Finite Stabilizer

by Tom Meyerovitch

Entropy 2016, 18(7), 263; https://doi.org/10.3390/e18070263 - 18 Jul 2016

Cited by 6 | Viewed by 3932

Abstract

We prove that, for a measure preserving action of a sofic group with positive sofic entropy, the stabilizer is finite on a set of positive measures. This extends the results of Weiss and Seward for amenable groups and free groups, respectively. It follows that the action of a sofic group on its subgroups by inner automorphisms has zero topological sofic entropy, and that a faithful action that has completely positive sofic entropy must be free. Full article

(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)

375 KiB

Open AccessFeature PaperArticle

Constant Slope Maps and the Vere-Jones Classification

by Jozef Bobok and Henk Bruin

Entropy 2016, 18(6), 234; https://doi.org/10.3390/e18060234 - 22 Jun 2016

Cited by 5 | Viewed by 4283

Abstract

We study continuous countably-piecewise monotone interval maps and formulate conditions under which these are conjugate to maps of constant slope, particularly when this slope is given by the topological entropy of the map. We confine our investigation to the Markov case and phrase [...] Read more.

(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)

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222 KiB

Open AccessArticle

On Extensions over Semigroups and Applications

by Wen Huang, Lei Jin and Xiangdong Ye

Entropy 2016, 18(6), 230; https://doi.org/10.3390/e18060230 - 15 Jun 2016

Cited by 3 | Viewed by 4413

Abstract

Applying a theorem according to Rhemtulla and Formanek, we partially solve an open problem raised by Hochman with an affirmative answer. Namely, we show that if G is a countable torsion-free locally nilpotent group that acts by homeomorphisms on X, and [...] Read more.

S \subset G

is a subsemigroup not containing the unit of G such that

f \in 〈 1, s f : s \in S 〉

for every

f \in C (X)

, then

(X, G)

has zero topological entropy. Full article

(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)

319 KiB

Open AccessFeature PaperArticle

Zero Entropy Is Generic

by Lewis Bowen

Entropy 2016, 18(6), 220; https://doi.org/10.3390/e18060220 - 4 Jun 2016

Cited by 7 | Viewed by 5207

Abstract

Dan Rudolph showed that for an amenable group, Γ, the generic measure-preserving action of Γ on a Lebesgue space has zero entropy. Here, this is extended to nonamenable groups. In fact, the proof shows that every action is a factor of a zero entropy action! This uses the strange phenomena that in the presence of nonamenability, entropy can increase under a factor map. The proof uses Seward’s recent generalization of Sinai’s Factor Theorem, the Gaboriau–Lyons result and my theorem that for every nonabelian free group, all Bernoulli shifts factor onto each other. Full article

(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)

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