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

<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>

Loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli

Foreman and Weiss [Measure preserving diffeomorphisms of the torus are unclassifiable. Preprint, 2020, arXiv:1705.04414] obtained an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphism, by using a functor $\mathcal {F}$ (see [Foreman and Weiss, From odometers to circular systems: a global structure theorem. J. Mod. Dyn.15 (2019), 345–423]) mapping odometer-based systems, $\mathcal {OB}$ , to circular systems, $\mathcal {CB}$ . This functor transfers the classification problem from $\mathcal {OB}$ to $\mathcal {CB}$ , and it preserves weakly mixing extensions, compact extensions, factor maps, the rank-one property, and certain types of isomorphisms. Thus it is natural to ask whether $\mathcal {F}$ preserves other dynamical properties. We show that $\mathcal {F}$ does not preserve the loosely Bernoulli property by providing positive and zero-entropy examples of loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli. We also construct a loosely Bernoulli circular system whose corresponding odometer-based system has zero entropy and is not loosely Bernoulli.

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel–Anosov–Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorphism of a new space that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms with positive finite entropy. Additionally, we show that such examples occur as minimal sets of volume preserving smooth diffeomorphisms of 4-dimensional manifolds.We construct an example of a minimal, weakly mixing and uniformly rigid homeomorphism of the pseudo-circle, and by our method we are also able to extend it to other one-dimensional hereditarily indecomposable continua, thereby producing the first examples of minimal, uniformly rigid and weakly mixing homeomorphisms in dimension 1. We also show that the examples we construct can be realized as invariant sets of smooth diffeomorphisms of a 4-manifold. Until now the only known examples of connected spaces that admit minimal, uniformly rigid and weakly mixing homeomorphisms were modifications of those given by Glasner and Maon in dimension at least 2.

Conceptions of Topological Transitivity on Symmetric Products

Let X be a topological space. For any positive integer n , we consider the n -fold symmetric product of X , ℱ n ( X ), consisting of all nonempty subsets of X with at most n points; and for a given function ƒ : X → X , we consider the induced functions ℱ n ( ƒ ): ℱ n ( X ) → ℱ n ( X ). Let M be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ + -transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal, I N, T T ++ , semi-open and irreducible. In this paper we study the relationship between the following statements: ƒ ∈ M and ℱ n ( ƒ ) ∈ M .

Rigidity, weak mixing, and recurrence in abelian groups

<p style='text-indent:20px;'>The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions extend to this setting:</p><p style='text-indent:20px;'>1. If <inline-formula><tex-math id="M2">\begin{document}$ (a_n) $\end{document}</tex-math></inline-formula> is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.</p><p style='text-indent:20px;'>2. There exists a sequence <inline-formula><tex-math id="M3">\begin{document}$ (r_n) $\end{document}</tex-math></inline-formula> such that every translate is both a rigidity sequence and a set of recurrence.</p><p style='text-indent:20px;'>The first of these results was shown for <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions by Adams [<xref ref-type="bibr" rid="b1">1</xref>], Fayad and Thouvenot [<xref ref-type="bibr" rid="b20">20</xref>], and Badea and Grivaux [<xref ref-type="bibr" rid="b2">2</xref>]. The latter was established in <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula> by Griesmer [<xref ref-type="bibr" rid="b23">23</xref>]. While techniques for handling <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.</p><p style='text-indent:20px;'>As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>, while others exhibit new phenomena.</p>

Transitivity in Fuzzy Hyperspaces

Given a metric space (X,d), we deal with a classical problem in the theory of hyperspaces: how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity) between a discrete dynamical system f:(X,d)→(X,d) and its natural extension to the hyperspace are related. In this context, we consider the Zadeh’s extension f^ of f to F(X), the family of all normal fuzzy sets on X, i.e., the hyperspace F(X) of all upper semicontinuous fuzzy sets on X with compact supports and non-empty levels and we endow F(X) with different metrics: the supremum metric d∞, the Skorokhod metric d0, the sendograph metric dS and the endograph metric dE. Among other things, the following results are presented: (1) If (X,d) is a metric space, then the following conditions are equivalent: (a) (X,f) is weakly mixing, (b) ((F(X),d∞),f^) is transitive, (c) ((F(X),d0),f^) is transitive and (d) ((F(X),dS)),f^) is transitive, (2) if f:(X,d)→(X,d) is a continuous function, then the following hold: (a) if ((F(X),dS),f^) is transitive, then ((F(X),dE),f^) is transitive, (b) if ((F(X),dS),f^) is transitive, then (X,f) is transitive; and (3) if (X,d) be a complete metric space, then the following conditions are equivalent: (a) (X×X,f×f) is point-transitive and (b) ((F(X),d0) is point-transitive.