Minimal systems and distributionally scrambled sets

Piotr Oprocha

Displaying similar documents to “Minimal systems and distributionally scrambled sets”

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...

On some properties of three-dimensional minimal sets in $ℝ^{4}$

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in $ℝ^{4}$ around a $𝕐$ -point and the existence of a point of particular type of a Mumford-Shah minimal set in $ℝ^{4}$ , which is very close to a $𝕋$ . This will give a local description of minimal sets of dimension 3 in $ℝ^{4}$ around a singular point and a property of Mumford-Shah minimal sets in $ℝ^{4}$ .

Zero-set property of o-minimal indefinitely Peano differentiable functions

Andreas Fischer (2008)

Annales Polonici Mathematici

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Given an o-minimal expansion ℳ of a real closed field R which is not polynomially bounded. Let $^{\infty}$ denote the definable indefinitely Peano differentiable functions. If we further assume that ℳ admits $^{\infty}$ cell decomposition, each definable closed subset A of Rⁿ is the zero-set of a $^{\infty}$ function f:Rⁿ → R. This implies $^{\infty}$ approximation of definable continuous functions and gluing of $^{\infty}$ functions defined on closed definable sets.

Interaction between cellularity of Alexandroff spaces and entropy of generalized shift maps

Fatemah Ayatollah Zadeh Shirazi, Sahar Karimzadeh Dolatabad, Sara Shamloo (2016)

Commentationes Mathematicae Universitatis Carolinae

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In the following text for a discrete finite nonempty set $K$ and a self-map $ϕ : X \to X$ we investigate interaction between different entropies of generalized shift $σ_{ϕ} : K^{X} \to K^{X}$ , ${(x_{α})}_{α \in X} \mapsto {(x_{ϕ (α)})}_{α \in X}$ and cellularities of some Alexandroff topologies on $X$ .

The generalized Toeplitz operators on the Fock space $F_{α}^{2}$

Chunxu Xu, Tao Yu (2024)

Czechoslovak Mathematical Journal

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Let $μ$ be a positive Borel measure on the complex plane $ℂ^{n}$ and let $j = (j_{1}, \dots, j_{n})$ with $j_{i} \in ℕ$ . We study the generalized Toeplitz operators $T_{μ}^{(j)}$ on the Fock space $F_{α}^{2}$ . We prove that $T_{μ}^{(j)}$ is bounded (or compact) on $F_{α}^{2}$ if and only if $μ$ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for $T_{μ}^{(j)}$ to be in the Schatten $p$ -class for $1 \leq p < \infty$ .

Topological dynamics of unordered Ramsey structures

Moritz Müller, András Pongrácz (2015)

Fundamenta Mathematicae

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We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M (G_{)}$ of the automorphism group $G$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of $M (G_{)}$ . We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M (G_{)}$ admits...

Renormings of $c_{0}$ and the minimal displacement problem

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this paper is to show that for every Banach space $(X, ∥ \cdot ∥)$ containing asymptotically isometric copy of the space $c_{0}$ there is a bounded, closed and convex set $C \subset X$ with the Chebyshev radius $r (C) = 1$ such that for every $k \geq 1$ there exists a $k$ -contractive mapping $T : C \to C$ with $∥ x - T x ∥ > 1 - 1 / k$ for any $x \in C$ .

Nonconventional limit theorems in averaging

Yuri Kifer (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider “nonconventional” averaging setup in the form $\frac{d X^{ε} (t)}{d t} = ε B (X^{ε} (t)$ , $𝛯 (q_{1} (t)), 𝛯 (q_{2} (t)), ..., 𝛯 (q_{ℓ} (t)))$ where $𝛯 (t)$ , $t \geq 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_{j} (t) = α_{j} t$ , $α_{1} l t; α_{2} l t; \dots l t; α_{k}$ and $q_{j}$ , $j = k + 1, ..., ℓ$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

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Given a bounded open set $Ω$ in $ℝ^{n}$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_{j}$ , we consider the quantity ${𝚖𝚊𝚡}_{j} λ (D_{j})$ where $λ (D_{j})$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_{j}$ . If we denote by $ℒ_{k} (Ω)$ the infimum over all the $k$ -partitions of ${𝚖𝚊𝚡}_{j} λ (D_{j})$ , a minimal $k$ -partition is then a partition which realizes the infimum. When $k = 2$ , we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$ ’s is non trivial and quite interesting. In this...

On open maps and related functions over the Salbany compactification

Mbekezeli Nxumalo (2024)

Archivum Mathematicum

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Given a topological space $X$ , let $𝒰 X$ and $η_{X} : X \to 𝒰 X$ denote, respectively, the Salbany compactification of $X$ and the compactification map called the Salbany map of $X$ . For every continuous function $f : X \to Y$ , there is a continuous function $𝒰 f : 𝒰 X \to 𝒰 Y$ , called the Salbany lift of $f$ , satisfying $(𝒰 f) \circ η_{X} = η_{Y} \circ f$ . If a continuous function $f : X \to Y$ has a stably compact codomain $Y$ , then there is a Salbany extension $F : 𝒰 X \to Y$ of $f$ , not necessarily unique, such that $F \circ η_{X} = f$ . In this paper, we give a condition on a space such that its Salbany map is open. In...

On preimages of ultrafilters in ZF

Horst Herrlich, Paul Howard, Kyriakos Keremedis (2016)

Commentationes Mathematicae Universitatis Carolinae

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We show that given infinite sets $X, Y$ and a function $f : X \to Y$ which is onto and $n$ -to-one for some $n \in ℕ$ , the preimage of any ultrafilter $ℱ$ of $Y$ under $f$ extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model $ℳ$ with a set of atoms $A$ and a finite-to-one onto function $f : A \to ω$ such that for each free ultrafilter of $ω$ its preimage under $f$ does not extend to an ultrafilter. In addition, we show that in $ℳ$ there exists an ultrafilter compact...

Product property for capacities in $ℂ^{N}$

Mirosław Baran, Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

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The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{ν} (E ₁ \times E ₂) = m i n (C_{ν ₁} (E ₁), C_{ν ₂} (E ₂))$ , where $E_{j}$ and $ν_{j}$ are respectively a compact set and a norm in $ℂ^{N_{j}}$ (j = 1,2), and ν is a norm in $ℂ^{N ₁ + N ₂}$ , ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of $ℂ^{N}$ , denote by C(E) the standard L-capacity and by $ω_{E}$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

$Σ_{s}$ -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any $Σ_{s}$ -product of at most $𝔠$ -many $L Σ (\leq ω)$ -spaces has the $L Σ (\leq ω)$ -property. This result generalizes some known results about $L Σ (\leq ω)$ -spaces. On the other hand, we prove that every $Σ_{s}$ -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every $Σ_{s}$ -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

On the birational gonalities of smooth curves

E. Ballico (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $C$ be a smooth curve of genus $g$ . For each positive integer $r$ the birational $r$ -gonality $s_{r} (C)$ of $C$ is the minimal integer $t$ such that there is $L \in {Pic}^{t} (C)$ with $h^{0} (C, L) = r + 1$ . Fix an integer $r \geq 3$ . In this paper we prove the existence of an integer $g_{r}$ such that for every integer $g \geq g_{r}$ there is a smooth curve $C$ of genus $g$ with $s_{r + 1} (C) / (r + 1) > s_{r} (C) / r$ , i.e. in the sequence of all birational gonalities of $C$ at least one of the slope inequalities fails.