scholarly journals An answer to Furstenberg’s problem on topological disjointness

2019 ◽  
Vol 40 (9) ◽  
pp. 2467-2481 ◽  
Author(s):  
WEN HUANG ◽  
SONG SHAO ◽  
XIANGDONG YE
Keyword(s):  
Open Subset ◽  
Dense Subset ◽  
Minimal System ◽  
Open Neighbourhood ◽  
Countable Dense Subset ◽  
Weakly Mixing ◽  
Transitive System

In this paper we give an answer to Furstenberg’s problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $y\in Y$ and any open neighbourhood $V$ of $y$, and for any non-empty open subset $U\subset X$, there is $x\in D\cap U$ such that $\{n\in \mathbb{Z}_{+}:T^{n}x\in U,S^{n}y\in V\}$ is syndetic. Some characterization for the general case is also given. By way of application we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^{n},T^{(n)})$ and $(X,T^{n})$ for any $n\in \mathbb{N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_{K})$ is disjoint from all minimal systems.

scholarly journals On $ n $-tuplewise IP-sensitivity and thick sensitivity

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jian Li ◽  
Yini Yang
Keyword(s):  
Dynamical System ◽  
Open Subset ◽  
Necessary Conditions ◽  
Minimal System ◽  
Topological Dynamical System ◽  
Minimal Points ◽  
Sufficient And Necessary Conditions ◽  
Weakly Mixing ◽  
Factor Maps

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M2">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> be a topological dynamical system and <inline-formula><tex-math id="M3">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>. We say that <inline-formula><tex-math id="M4">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive (resp. <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive) if there exists a constant <inline-formula><tex-math id="M7">\begin{document}$ \delta&gt;0 $\end{document}</tex-math></inline-formula> with the property that for each non-empty open subset <inline-formula><tex-math id="M8">\begin{document}$ U $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M9">\begin{document}$ X $\end{document}</tex-math></inline-formula>, there exist <inline-formula><tex-math id="M10">\begin{document}$ x_1,x_2,\dotsc,x_n\in U $\end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Bigl\{k\in \mathbb{N}\colon \min\limits_{1\le i&lt;j\le n}d(T^k x_i,T^k x_j)&gt;\delta\Bigr\} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is an IP-set (resp. a thick set).</p><p style='text-indent:20px;'>We obtain several sufficient and necessary conditions of a dynamical system to be <inline-formula><tex-math id="M11">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive or <inline-formula><tex-math id="M12">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise IP-sensitive for all <inline-formula><tex-math id="M14">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>, while it is <inline-formula><tex-math id="M15">\begin{document}$ n $\end{document}</tex-math></inline-formula>-tuplewise thickly sensitive if and only if it has at least <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula> minimal points. We characterize two kinds of sensitivity by considering some kind of factor maps. We introduce the opposite side of pairwise IP-sensitivity and pairwise thick sensitivity, named (almost) pairwise IP<inline-formula><tex-math id="M17">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuity and (almost) pairwise syndetic equicontinuity, and obtain dichotomies results for them. In particular, we show that a minimal system is distal if and only if it is pairwise IP<inline-formula><tex-math id="M18">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuous. We show that every minimal system admits a maximal almost pairwise IP<inline-formula><tex-math id="M19">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-equicontinuous factor and admits a maximal pairwise syndetic equicontinuous factor, and characterize them by the factor maps to their maximal distal factors.</p>

scholarly journals Point transitivity, -transitivity and multi-minimality

2014 ◽  
Vol 35 (5) ◽  
pp. 1423-1442 ◽  
Author(s):  
ZHIJING CHEN ◽  
JIAN LI ◽  
JIE LÜ
Keyword(s):  
Dynamical System ◽  
Open Subset ◽  
Topological Dynamical System ◽  
Property A ◽  
Weak Mixing ◽  
Furstenberg Family ◽  
Strongly Mixing ◽  
Transitive Point ◽  
Weakly Mixing

Let $(X,f)$ be a topological dynamical system and ${\mathcal{F}}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an ${\mathcal{F}}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}:f^{n}(x)\in U\}$, is in ${\mathcal{F}}$; the system $(X,f)$ is called ${\mathcal{F}}$-point transitive if there exists some ${\mathcal{F}}$-transitive point. In this paper, we first discuss the connection between ${\mathcal{F}}$-point transitivity and ${\mathcal{F}}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by ${\mathcal{F}}$-point transitivity, completing results in [Transitive points via Furstenberg family. Topology Appl. 158 (2011), 2221–2231]. We also show that multi-transitivity, ${\rm\Delta}$-transitivity and multi-minimality can be characterized by ${\mathcal{F}}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates. Ergod. Th. & Dynam. Sys. 32 (2012), 1661–1672].

scholarly journals FRÉCHET INTERMEDIATE DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS ON ASPLUND SPACES

2009 ◽  
Vol 79 (2) ◽  
pp. 309-317 ◽  
Author(s):  
J. R. GILES
Keyword(s):  
Large Class ◽  
Open Subset ◽  
Lipschitz Function ◽  
Dense Subset ◽  
Lipschitz Functions ◽  
Nonempty Open Subset ◽  
Asplund Space ◽  
Asplund Spaces ◽  
Fréchet Differentiable

AbstractThe deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

scholarly journals On weak mixing, minimality and weak disjointness of all iterates

2011 ◽  
Vol 32 (5) ◽  
pp. 1661-1672 ◽  
Author(s):  
DOMINIK KWIETNIAK ◽  
PIOTR OPROCHA
Keyword(s):  
Topological Dynamical System ◽  
Multiple Recurrence ◽  
Weak Mixing ◽  
Compact Metric Space ◽  
Open Questions ◽  
Weakly Mixing ◽  
Recurrence Theorem ◽  
Topological Weak Mixing ◽  
Mixing Property ◽  
Transitive System

AbstractThis article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map f×f2×⋯×fm, where f:X→X is a topological dynamical system on a compact metric space. The theorem stating that a weakly mixing and strongly transitive system is Δ-transitive is extended to a non-invertible case with a simple proof. Two examples are constructed, answering the questions posed by Moothathu [Diagonal points having dense orbit. Colloq. Math. 120(1) (2010), 127–138]. The first one is a multi-transitive non-weakly mixing system, and the second one is a weakly mixing non-multi-transitive system. The examples are special spacing shifts. The latter shows that the assumption of minimality in the multiple recurrence theorem cannot be replaced by weak mixing.

scholarly journals A disjointly tight irresolvable space

2020 ◽  
Vol 21 (2) ◽  
pp. 326
Author(s):  
Angelo Bella ◽  
Michael Hrusak
Keyword(s):  
Dense Subset ◽  
Short Note ◽  
Completely Regular ◽  
Countable Dense Subset

<p>In this short note we prove the existence (in ZFC) of a completely regular countable disjointly tight irresolvable space by showing that every sub-maximal countable dense subset of 2c is disjointly tight.</p>

scholarly journals On super-exponential divergence of periodic points for partially hyperbolic systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiaolong Li ◽  
Katsutoshi Shinohara
Keyword(s):  
Growth Rate ◽  
Lower Limit ◽  
Open Subset ◽  
Exponential Growth ◽  
Dense Subset ◽  
Hyperbolic Systems ◽  
Periodic Points ◽  
Residual Subset ◽  
Partially Hyperbolic ◽  
Smooth Closed Manifold

<p style='text-indent:20px;'>We say that a diffeomorphism <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> is super-exponentially divergent if for every <inline-formula><tex-math id="M2">\begin{document}$ b&gt;1 $\end{document}</tex-math></inline-formula> the lower limit of <inline-formula><tex-math id="M3">\begin{document}$ \#\mbox{Per}_n(f)/b^n $\end{document}</tex-math></inline-formula> diverges to infinity, where <inline-formula><tex-math id="M4">\begin{document}$ \mbox{Per}_n(f) $\end{document}</tex-math></inline-formula> is the set of all periodic points of <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> with period <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>. This property is stronger than the usual super-exponential growth of the number of periodic points. We show that for any <inline-formula><tex-math id="M7">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional smooth closed manifold <inline-formula><tex-math id="M8">\begin{document}$ M $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M9">\begin{document}$ n\ge 3 $\end{document}</tex-math></inline-formula>, there exists a non-empty open subset <inline-formula><tex-math id="M10">\begin{document}$ \mathcal{O} $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M11">\begin{document}$ \mbox{Diff}^1(M) $\end{document}</tex-math></inline-formula> such that diffeomorphisms with super-exponentially divergent property form a dense subset of <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{O} $\end{document}</tex-math></inline-formula> in the <inline-formula><tex-math id="M13">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>-topology. A relevant result about the growth rate of the lower limit of the number of periodic points for diffeomorphisms in a <inline-formula><tex-math id="M14">\begin{document}$ C^r $\end{document}</tex-math></inline-formula>-residual subset of <inline-formula><tex-math id="M15">\begin{document}$ \mbox{Diff}^r(M)\ (1\le r\le \infty) $\end{document}</tex-math></inline-formula> is also shown.</p>

scholarly journals Some properties of the containing spaces and saturated classes of spaces

2003 ◽  
Vol 4 (2) ◽  
pp. 487 ◽  
Author(s):  
S.D. Iliadis
Keyword(s):  
Open Subset ◽  
Closed Subset ◽  
Dense Subset ◽  
Simple Condition ◽  
Specific Subset ◽  
Universal Element ◽  
The Given ◽  
Ordered Pairs

<p>Subjects of this paper are: (a) containing spaces constructed in [2] for an indexed collection S of subsets, (b) classes consisting of ordered pairs (Q,X), where Q is a subset of a space X, which are called classes of subsets, and (c) the notion of universality in such classes.</p> <p>We show that if T is a containing space constructed for an indexed collection S of spaces and for every X ϵ S, Q<sup>X</sup> is a subset of X, then the corresponding containing space TI<sub>Q</sub> constructed for the indexed collection Q ={Q<sup>X</sup> : X ϵ S} of spaces, under a simple condition, can be considered as a specific subset of T. We prove some “commutative” properties of these specific subsets.</p> <p>For classes of subsets we introduce the notion of a (properly) universal element and define the notion of a (complete) saturated class of subsets. Such a class is “saturated” by (properly) universal elements. We prove that the intersection of (complete) saturated classes of subsets is also a (complete) saturated class.</p> <p>We consider the following classes of subsets: (a) IP(Cl), (b) IP(Op), and (c) IP(n.dense) consisting of all pairs (Q;X) such that: (a) Q is a closed subset of X, (b) Q is an open subset of X, and (c) Q is a never dense subset of X, respectively. We prove that the classes IP(Cl) and IP(Op) are complete saturated and the class IP(n.dense) is saturated. Saturated classes of subsets are convenient to use for the construction of new saturated classes by the given ones.</p>

Algebras of Holomorphic Functions in Ringed Spaces, I

1969 ◽  
Vol 21 ◽  
pp. 1281-1292
Author(s):  
Maxwell E. Shauck
Keyword(s):  
Open Subset ◽  
Hausdorff Space ◽  
Holomorphic Functions ◽  
Continuous Functions ◽  
Open Neighbourhood ◽  
Continuous Complex ◽  
Algebras Of Holomorphic Functions ◽  
Complex Valued

A pair () is a ringed space if it is a subsheaf of rings with 1 of the sheaf of germs of continuous functions on X. If U is an open subset of X, we denote the set of sections over U relative to by . If , then implies that there exists some open neighbourhood V of u, V ⊂ U, and some g continuous on V such that the germ of g at u, ug is ϕ(u). Now we define ϕ(u) (u) to be g(u) and in this way we obtain, in a unique fashion, a continuous complex-valued function on U. The collection of all such functions for a given set is denoted by and is called the -holomorphic functions on U.THEOREM. Let X be a locally connected Hausdorff space and () a ringed space.

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