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Sequence entropy pairs and complexity pairs for a measure

Wen Huang, Alejandro Maass, Xiangdong Ye (2004)

Annales de l’institut Fourier

In this paper we explore topological factors in between the Kronecker factor and the maximal equicontinuous factor of a system. For this purpose we introduce the concept of sequence entropy n -tuple for a measure and we show that the set of sequence entropy tuples for a measure is contained in the set of topological sequence entropy tuples [H- Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we introduce a weak notion and a strong notion of complexity pair for a...

Symbolic extensions in intermediate smoothness on surfaces

David Burguet (2012)

Annales scientifiques de l'École Normale Supérieure

We prove that 𝒞 r maps with r > 1 on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].

The entropy conjecture for diffeomorphisms away from tangencies

Gang Liao, Marcelo Viana, Jiagang Yang (2013)

Journal of the European Mathematical Society

We prove that every C 1 diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive.

The entropy of algebraic actions of countable torsion-free abelian groups

Richard Miles (2008)

Fundamenta Mathematicae

This paper is concerned with the entropy of an action of a countable torsion-free abelian group G by continuous automorphisms of a compact abelian group X. A formula is obtained that expresses the entropy in terms of the Mahler measure of a greatest common divisor, complementing earlier work by Einsiedler, Lind, Schmidt and Ward. This leads to a uniform method for calculating entropy whenever G is free. In cases where these methods do not apply, a possible entropy formula is conjectured. The entropy...

The recurrence dimension for piecewise monotonic maps of the interval

Franz Hofbauer (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval [ 0 , 1 ] , giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure...

The set of recurrent points of a continuous self-map on compact metric spaces and strong chaos

Lidong Wang, Gongfu Liao, Zhizhi Chen, Xiaodong Duan (2003)

Annales Polonici Mathematici

We discuss the existence of an uncountable strongly chaotic set of a continuous self-map on a compact metric space. It is proved that if a continuous self-map on a compact metric space has a regular shift invariant set then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.

The topological entropy versus level sets for interval maps

Jozef Bobok (2002)

Studia Mathematica

We answer affirmatively Coven's question [PC]: Suppose f: I → I is a continuous function of the interval such that every point has at least two preimages. Is it true that the topological entropy of f is greater than or equal to log 2?

The topological entropy versus level sets for interval maps (part II)

Jozef Bobok (2005)

Studia Mathematica

Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) c a r d f - 1 ( y ) ≥ 2 for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that c a r d f - 1 ( y ) ≥ m for every y ∈ [a,b]∖L. We show that the topological entropy of f is greater than or equal to log m. This generalizes our previous result for m = 2.

Time weighted entropies

Jörg Schmeling (2000)

Colloquium Mathematicae

For invertible transformations we introduce various notions of topological entropy. For compact invariant sets these notions are all the same and equal the usual topological entropy. We show that for non-invariant sets these notions are different. They can be used to detect the direction in time in which the system evolves to highest complexity.

Topological entropy and differential equations

Ján Andres, Pavel Ludvík (2023)

Archivum Mathematicum

On the background of a brief survey panorama of results on the topic in the title, one new theorem is presented concerning a positive topological entropy (i.e. topological chaos) for the impulsive differential equations on the Cartesian product of compact intervals, which is positively invariant under the composition of the associated Poincaré translation operator with a multivalued upper semicontinuous impulsive mapping.

Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Sergiĭ Kolyada, Michał Misiurewicz, L’ubomír Snoha (1999)

Fundamenta Mathematicae

The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces ( X i ) i = 1 ∞ and a sequence of continuous maps ( f i ) i = 1 ∞ , f i : X i → X i + 1 , is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of f n ○ . . . ○ f 2 ○ f 1 . As an application we construct a large class of smooth triangular maps of the square of type 2 ∞ and positive...

Topological entropy on zero-dimensional spaces

Jozef Bobok, Ondřej Zindulka (1999)

Fundamenta Mathematicae

Let X be an uncountable compact metrizable space of topological dimension zero. Given any a ∈[0,∞] there is a homeomorphism on X whose topological entropy is a.

Currently displaying 61 – 80 of 86