Topological size of scrambled sets

François Blanchard; Wen Huang; L'ubomír Snoha

Displaying similar documents to “Topological size of scrambled sets”

Topological sequence entropy for maps of the circle

Roman Hric (2000)

Commentationes Mathematicae Universitatis Carolinae

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A continuous map $f$ of the interval is chaotic iff there is an increasing sequence of nonnegative integers $T$ such that the topological sequence entropy of $f$ relative to $T$ , $h_{T} (f)$ , is positive ([FS]). On the other hand, for any increasing sequence of nonnegative integers $T$ there is a chaotic map $f$ of the interval such that $h_{T} (f) = 0$ ([H]). We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning topological sequence entropy for maps of general compact...

Defining complete and observable chaos

Víctor Jiménez López (1996)

Annales Polonici Mathematici

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For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which $l i m i n f_{n \to \infty} | f^{n} (x) - f^{n} (y) | = 0$ and $l i m s u p_{n \to \infty} | f^{n} (x) - f^{n} (y) | > 0$ . We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible...

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

Jozef Bobok (2005)

Studia Mathematica

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Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) $c a r d f^{- 1} (y) \geq 2$ for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that $c a r d f^{- 1} (y) \geq m$ 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.

Jumps of entropy for $C^{r}$ interval maps

David Burguet (2015)

Fundamenta Mathematicae

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We study the jumps of topological entropy for $C^{r}$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^{r} ([0, 1])$ with $h_{t o p} (f) > (l o g ⁺ | | f^{'} {| |}_{\infty}) / r$ . To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^{r}$ interval maps.

Minimal systems and distributionally scrambled sets

Piotr Oprocha (2012)

Bulletin de la Société Mathématique de France

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In this paper we investigate numerous constructions of minimal systems from the point of view of $(ℱ_{1}, ℱ_{2})$ -chaos (but most of our results concern the particular cases of distributional chaos of type $1$ and $2$ ). We consider standard classes of systems, such as Toeplitz flows, Grillenberger $K$ -systems or Blanchard-Kwiatkowski extensions of the Chacón flow, proving that all of them are DC2. An example of DC1 minimal system with positive topological entropy is also introduced. The above mentioned results...

Topology and measure of buried points in Julia sets

Clinton P. Curry, John C. Mayer, E. D. Tymchatyn (2013)

Fundamenta Mathematicae

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It is well-known that the set of buried points of a Julia set of a rational function (also called the residual Julia set) is topologically “fat” in the sense that it is a dense $G_{δ}$ if it is non-empty. We show that it is, in many cases, a full-measure subset of the Julia set with respect to conformal measure and the measure of maximal entropy. We also address Hausdorff dimension of buried points in the same cases, and discuss connectivity and topological dimension of the set of buried points....

Topological disjointness from entropy zero systems

Wen Huang, Kyewon Koh Park, Xiangdong Ye (2007)

Bulletin de la Société Mathématique de France

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The properties of topological dynamical systems $(X, T)$ which are disjoint from all minimal systems of zero entropy, $ℳ_{0}$ , are investigated. Unlike the measurable case, it is known that topological $K$ -systems make up a proper subset of the systems which are disjoint from $ℳ_{0}$ . We show that $(X, T)$ has an invariant measure with full support, and if in addition $(X, T)$ is transitive, then $(X, T)$ is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists...

Semiconjugacy to a map of a constant slope

Jozef Bobok (2012)

Studia Mathematica

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It is well known that any continuous piecewise monotone interval map f with positive topological entropy $h_{t o p} (f)$ is semiconjugate to some piecewise affine map with constant slope $e^{h_{t o p} (f)}$ . We prove this result for a class of Markov countably piecewise monotone continuous interval maps.

Symbolic extensions in intermediate smoothness on surfaces

David Burguet (2012)

Annales scientifiques de l'École Normale Supérieure

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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].

Commutativity and non-commutativity of topological sequence entropy

Francisco Balibrea, Jose Salvador Cánovas Peña, Víctor Jiménez López (1999)

Annales de l'institut Fourier

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In this paper we study the commutativity property for topological sequence entropy. We prove that if $X$ is a compact metric space and $f, g : X \to X$ are continuous maps then $h_{A} (f \circ g) = h_{A} (g \circ f)$ for every increasing sequence $A$ if $X = [0, 1]$ , and construct a counterexample for the general case. In the interim, we also show that the equality $h_{A} (f) = h_{A} (f |_{\cap_{n \geq 0} f^{n} (X)})$ is true if $X = [0, 1]$ but does not necessarily hold if $X$ is an arbitrary compact metric space.

On the joint entropy of $d$ -wise-independent variables

Dmitry Gavinsky, Pavel Pudlák (2016)

Commentationes Mathematicae Universitatis Carolinae

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How low can the joint entropy of $n$ $d$ -wise independent (for $d \geq 2$ ) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than $p$ , for $p < 1$ )? This question has been posed and partially answered in a recent work of Babai [Entropy versus pairwise independence (preliminary version), http://people.cs.uchicago.edu/ laci/papers/13augEntropy.pdf, 2013]. In this paper we improve some...

Large sets of integers and hierarchy of mixing properties of measure preserving systems

Vitaly Bergelson, Tomasz Downarowicz (2008)

Colloquium Mathematicae

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We consider a hierarchy of notions of largeness for subsets of ℤ (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βℤ to establish connections between various notions of largeness and apply those results to the study of the sets $R_{A, B}^{ε} = n \in ℤ : μ (A \cap T ⁿ B) > μ (A) μ (B) - ε$ of times of “fat intersection”. Among other things we show that the sets $R_{A, B}^{ε}$ allow one...

ε-Entropy and moduli of smoothness in $L^{p}$ -spaces

A. Kamont (1992)

Studia Mathematica

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The asymptotic behaviour of ε-entropy of classes of Lipschitz functions in $L^{p} (^{d})$ is obtained. Moreover, the asymptotics of ε-entropy of classes of Lipschitz functions in $L^{p} (ℝ^{d})$ whose tail function decreases as $O (λ^{- γ})$ is obtained. In case p = 1 the relation between the ε-entropy of a given class of probability densities on $ℝ^{d}$ and the minimax risk for that class is discussed.

The $n$ -centre problem of celestial mechanics for large energies

Andreas Knauf (2002)

Journal of the European Mathematical Society

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We consider the classical three-dimensional motion in a potential which is the sum of $n$ attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the $n$ centres, we find a universal behaviour for all energies $E$ above a positive threshold. Whereas for $n = 1$ there are no bounded orbits, and for $n = 2$ there is just one closed orbit, for $n \geq 3$ the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of...

Orders of accumulation of entropy

David Burguet, Kevin McGoff (2012)

Fundamenta Mathematicae

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For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structures and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if M is a metrizable Choquet...

On groups of essential values of topological cylinder cocycles over minimal rotations

Mieczysław K. Mentzen (2003)

Colloquium Mathematicae

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A theory of essential values of cocycles over minimal rotations with values in locally compact Abelian groups, especially $ℝ^{m}$ , is developed. Criteria for such a cocycle to be conservative are given. The group of essential values of a cocycle is described.

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Bernd Carl, David E. Edmunds (2003)

Studia Mathematica

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For a precompact subset K of a Hilbert space we prove the following inequalities: $n^{1 / 2} c ₙ (c o v (K)) \leq c_{K} (1 + \sum_{k = 1}^{ⁿ} k^{- 1 / 2} e_{k} (K))$ , n ∈ ℕ, and $k^{1 / 2} c_{k + n} (c o v (K)) \leq c [l o g^{1 / 2} (n + 1) ε ₙ (K) + \sum_{j = n + 1}^{\infty} ε_{j} (K) / (j l o g^{1 / 2} (j + 1))]$ , k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and $ε_{k} (K)$ and $e_{k} (K)$ denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K)...

Diagonal points having dense orbit

T. K. Subrahmonian Moothathu (2010)

Colloquium Mathematicae

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Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f \times f ² \times \dots \times f^{m} : X^{m} \to X^{m}$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^{m}$ under the action of $f \times f ² \times \dots \times f^{m}$ . We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying...

Maximal distributional chaos of weighted shift operators on Köthe sequence spaces

Xinxing Wu (2014)

Czechoslovak Mathematical Journal

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During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator $B_{w}^{n} : λ_{p} (A) \to λ_{p} (A)$ defined on the Köthe sequence space $λ_{p} (A)$ exhibits distributional $ϵ$ -chaos for any $0 < ϵ < diam λ_{p} (A)$ and any $n \in ℕ$ is obtained. Under this assumption, the principal measure of $B_{w}^{n}$ is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional $ϵ$ -chaos for any...