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In this paper, we study the shift on the space of uniformly bounded continuous functions band-limited in a given compact interval with the standard topology of tempered distributions. We give a constructive proof of the existence of minimal subsystems with any given mean dimension strictly less than twice its band-width. A version of real-valued function spaces is considered as well. [ABSTRACT FROM AUTHOR]

While studying gradient dynamical systems, Morse introduced the idea of encoding the qualitative behavior of a dynamical system into a graph. Smale later refined Morse's idea and extended it to Axiom-A diffeomorphisms on manifolds. In Smale's vision, nodes are indecomposable closed invariant subsets of the non-wandering set with a dense orbit and there is an edge from node M to node N (we say that N is downstream from M) if the unstable manifold of M intersects the stable manifold of N. Since then, the decomposition of the non-wandering set was studied in many other settings, while the edges component of Smale's construction has been often overlooked. In the same years, more sophisticated generalizations of the non-wandering set, introduced by Birkhoff in 1920s, were elaborated first by Auslander in early 1960s, by Conley in early 1970s and later by Easton and other authors. In our language, each of these generalizations involves the introduction of a closed and transitive extension of the prolongational relation, that is closed but not transitive. In the present article, we develop a theory that generalizes at the same time both these lines of research. We study the general properties of closed transitive relations (which we call streams) containing the space of orbits of a discrete-time or continuous-time semi-flow and we argue that these relations play a central role in the qualitative study of dynamical systems. All most studied concepts of recurrence currently in literature can be defined in terms of our streams. Finally, we show how to associate to each stream a graph encoding its qualitative properties. Our main general result is that each stream of a semi-flow with "compact dynamics" has a connected graph. The range of semi-flows covered by our theorem goes from 1-dimensional discrete-time systems like the logistic map up to infinite-dimensional continuous-time systems like the semi-flow of quasilinear parabolic reaction–diffusion partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we studied the metric mean dimension in Feldman–Katok (FK for short) metric. We introduced the notions of FK-Bowen metric mean dimension and FK-Packing metric mean dimension on subsets. And we established two variational principles. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a combinatorial proof approach for a theorem regarding the upper Banach density of a subset S of N and present a finite IP-set revision of the theorem originally established by Li, Tu, and Ye. [ABSTRACT FROM AUTHOR]

We study the dynamical properties of mean equicontinuity, mean sensitivity, tracking property, and periodic points set in the hyperspace of uniform space. Let (Y,ξ) be uniform space, T:Y→Y be uniformly continuous and (C(Y),Cξ) be hyperspace of (Y,ξ).Then, we can get some conclusions: (a) T is mean equicontinous if and only if the induced map CT is mean equicontinous; (b) if the induced map CT is mean sensitive, then T is mean sensitive; (c) the induced map CT has tracking property implying that T has tracking property; (d) P(T) is dense in Y implying that P(CT) is dense in C(Y). In addition, we also study dynamical property of (G,h)− tracking property in metric G− space, and prove that if the map T has (G,h)− tracking property, then, for any k>1, the map Tk has (G,h)− tracking property.

Mean dimension is a topological invariant of dynamical systems, which originates with Mikhail Gromov in 1999 and which was studied with deep applications around 2000 by Elon Lindenstrauss and Benjamin Weiss within the framework of amenable group actions. Let a countable discrete amenable group G act continuously on compact metrizable spaces X and Y. Consider the product action of G on the product space X × Y . The product inequality for mean dimension is well known: mdim (X × Y , G) ≤ mdim (X , G) + mdim (Y , G) , while it was unknown for a long time if the product inequality could be an equality. In 2019, Masaki Tsukamoto constructed the first example of two different continuous actions of G on compact metrizable spaces X and Y, respectively, such that the product inequality becomes strict. However, there is still one longstanding problem which remains open in this direction, asking if there exists a continuous action of G on some compact metrizable space X such that mdim (X × X , G) < 2 · mdim (X , G) . We solve this problem. Somewhat surprisingly, we prove, in contrast to (topological) dimension theory, a rather satisfactory theorem: If an infinite (countable discrete) amenable group G acts continuously on a compact metrizable space X, then we have mdim (X n , G) = n · mdim (X , G) , for any positive integer n. Our product formula for mean dimension, together with the example and inequality (stated previously), eventually allows mean dimension of product actions to be fully understood. [ABSTRACT FROM AUTHOR]

Our aim is to find some new links between linear (circular) orderability of groups and topological dynamics. We suggest natural analogs of the concept of algebraic orderability for topological groups involving order-preserving actions on compact spaces and the corresponding enveloping semigroups in the sense of R. Ellis. This approach leads to several natural questions. Some of them might be useful also for discrete (countable) orderable groups. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of GG-Lipschitz tracking property and GG-asymptotic tracking property in metric GG-space and obtain the equivalent conditions of GG-asymptotic tracking property in metric GG-space. In addition, we prove that the self-map ff has the GG-Lipschitz tracking property if and only if the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property in the inverse limit space under the topological group action. These results generalize the corresponding results in [Proc. Amer. Math. Soc. 115 (1992), 573–580].

First, we give the concepts of G-sequence shadowing property, G-equicontinuity and G-regularly recurrent point. Second, we study their dynamical properties in the inverse limit space under group action. The following results are obtained. (1) The self-mapping ff has the G-sequence shadowing property if and only if the shift mapping σ\sigma has the G¯\overline{G}-sequence shadowing property; (2) The self-mapping ff is G-equicontinuous if and only if the shift mapping σ\sigma is G¯\overline{G}-equicontinuous; (3) RRG¯(σ)=lim←(RRG(f),f)R{R}_{\overline{G}}\left(\sigma )=\underleftarrow{\mathrm{lim}}\left(R{R}_{G}(f),f). These conclusions make up for the lack of theory in the inverse limit space under group action.

We show that, given a compact minimal system $(X,g)$ and an element h of the topological full group $\tau [g]$ of g, the infinite orbits of h admit a locally constant orientation with respect to the orbits of g. We use this to obtain a clopen partition of $(X,G)$ into minimal and periodic parts, where G is any virtually polycyclic subgroup of $\tau [g]$. We also use the orientation of orbits to give a refinement of the index map and to describe the role in $\tau [g]$ of the submonoid generated by the induced transformations of g. Finally, we consider the problem, given a homeomorphism h of the Cantor space X, of determining whether or not there exists a minimal homeomorphism g of X such that $h \in \tau [g]$. [ABSTRACT FROM AUTHOR]

For a dynamical system (X, f), the notion of topological transitivity has been studied by some researchers. There are several definitions of this property, and it is part of the folklore of dynamical systems that under some hypotheses, they are equivalent. In this paper, our aim is to introduce and study some properties of topological transitivity in pointfree topology, for which we first need to define in a way what makes them conservative extensions of topological transitivity defined by G.D. Birkhoff. We describe the way the different properties are related to each other in pointfree topology. [ABSTRACT FROM AUTHOR]

We introduce various types of n-sensitivity, such as multi-n-sensitivity, strong n-sensitivity, syndetic n-sensitivity, thick n-sensitivity and thickly syndetic n-sensitivity in general semiflows, and provide several sufficient conditions under which a semiflow possesses such notions. We prove that multi-n-sensitivity, thick n-sensitivity and thickly syndetic n-sensitivity are all equivalent for an almost open P-M-semiflow whose acting semigroup is discrete and abelian. We also discuss the relationships among these notions, and provide examples wherever implications are not true. [ABSTRACT FROM AUTHOR]

We consider the (three parameters) family of nonlinear mappings F a , b , c (x , y) = (a x + b y , c y − b y − a x) where a, b, c are positive real numbers. F 1 , 1 , 1 is the classical Hénon-Devaney map [1, 2, 6]. For a large open region of parameters, we exhibit invariantCantor sets embedded in the plane with two hyperbolic fixed points of saddle type. [ABSTRACT FROM AUTHOR]

In this work, we look at the extension of classical discrete dynamical system to multidimensional discrete-time dynamical system by characterizing chaos notions on Z d -action. The Z d -action on a space X has been defined in a very general manner, and therefore we introduce a Z d -action on X which is induced by a continuous map, f : Z × X → X and denotes it as T f : Z d × X → X . Basically, we wish to relate the behavior of origin discrete dynamical systems (X, f) and its induced multidimensional discrete-time (X , T f) . The chaotic behaviors that we emphasized are the transitivity and dense periodicity property. Analogues to these chaos notions, we consider k-type transitivity and k-type dense periodicity property in the multidimensional discrete-time dynamical system. In the process, we obtain some conditions on (X , T f) under which the chaotic behavior of (X , T f) is inherited from the original dynamical system (X, f). The conditions varies whenever f is open, totally transitive or mixing. Some examples are given to illustrate these conditions. [ABSTRACT FROM AUTHOR]

In this paper we formulate the notions of upper metric mean dimension and upper measure-theoretic mean dimension for amenable group actions. In particular, a different version of variational principle for amenable group actions is presented by replacing the condition on the space in Velozo and Velozo (Rate distortion theory, metric mean dimension and measure theoretic entropy. Preprint ArXiV: 1707.05762 [Math.DS], 2017) by the conditions on the systems. At last, we discuss the relation between upper metric mean dimensions and pseudo-orbits. [ABSTRACT FROM AUTHOR]

In the following text we introduce specification property (stroboscopical property) for dynamical systems on uniform space. We focus on two classes of dynamical systems: generalized shifts and dynamical systems with Alexandroff compactification of a discrete space as phase space. We prove that for a discrete finite topological space X with at least two elements, a nonempty set Γ and a self-map φ : Γ →Γ the generalized shift dynamical system (XΓ,σφ): has (almost) weak specification property if and only if φ : Γ →Γ does not have any periodic point, has (uniform) stroboscopical property if and only if φ : Γ →Γ is one-to-one. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the topological pressures of finitely generated free semigroup actions on a compact metric space. By means of a factor map, we give an inequality relation for two topological pressures with a factor map of free semigroup actions. We establish a formula for two topological pressures with a factor map of free semigroup actions. [ABSTRACT FROM AUTHOR]

The Kechris–Pestov–Todorčević correspondence (KPT-correspondence for short) is a surprising correspondence between model theory, combinatorics and topological dynamics. In this paper we present a categorical re-interpretation of (a part of) the KPT-correspondence with the aim of proving a dual statement. Our strategy is to take a "direct" result and then analyze the necessary infrastructure that makes the result true by providing a purely categorical proof of the categorical version of the result. We can then capitalize on the Duality Principle to obtain the dual statements almost for free. We believe that the dual version of the KPT-correspondence can not only provide the new insights into the interplay of combinatorial, model-theoretic and topological phenomena this correspondence binds together, but also explores the limits to which categorical treatment of combinatorial phenomena can take us. [ABSTRACT FROM AUTHOR]

In this paper, we study the noncommutative functional equation h (λ z) - λ h (z) = g (z) , z ∈ C and we give a new perspective from this equation to obtain a completely dynamical classification for complex matrices. Coarsely speaking, there are four different types: 0 , 1 2 , 2 and e i 2 π θ with θ ∈ [ 0 , 1 2 ] for diagonal matrices and Jordan matrices, respectively. Moreover, we obtain that for a complex matrix A, if its eigenvalues are 0 < | λ i | ≠ 1 , where 1 ≤ i ≤ rank (A) , then A is topologically conjugate to a diagonal matrix on C rank (A) . [ABSTRACT FROM AUTHOR]

Supermixing and hypermixing strongly continuous semigroups are introduced in this paper. It is proved that supermixing preserves under quasiconjugacy. Moreover, it is established that if a strongly continuous semigroup is supermixing(hypermixing), then any discretization of it, is supermixing(hypermixing). Also, it is proved that supermixing(hypermixing) of (Tt)t ≥ 0 implies that any of Tt's is a supermixing(hypermixing). Some sufficient conditions for supermixing and hypermixing are stated that based on dense sets and kernels of operators of semigroup. It is proved that the supermixing(hypermixing) of two strongly continuous semigroups imply that their direct sum is supermixing(hypermixing). Also, supermixing(hypermixing) of the direct sum of finite strongly continuous semigroups indicates that any of these semigroups is supermixing(hypermixing). Outcomes of this study have shown that extension of the concepts of supermixing and hypermixing to strongly continuous semigroups leads to get access to a proper subset of the set of hypercyclic semigroups and reach valuable results about them. [ABSTRACT FROM AUTHOR]

We construct a multiply Xiong chaotic set with full Hausdorff dimension everywhere that is contained in some multiply proximal cell for the full shift over finite symbols and the Gauss system, respectively. [ABSTRACT FROM AUTHOR]

A nilspace system is a generalization of a nilsystem, consisting of a compact nilspace $\text{X}$ equipped with a group of nilspace translations acting on $\text{X}$. Nilspace systems appear in different guises in several recent works and this motivates the study of these systems per se as well as their relation to more classical types of systems. In this paper we study morphisms of nilspace systems, i.e., nilspace morphisms with the additional property of being consistent with the actions of the given translations. A nilspace morphism does not necessarily have this property, but one of our main results shows that it factors through some other morphism which does have the property. As an application we obtain a strengthening of the inverse limit theorem for compact nilspaces, valid for nilspace systems. This is used in work of the first- and third-named authors to generalize the celebrated structure theorem of Host and Kra on characteristic factors. [ABSTRACT FROM AUTHOR]

In this paper, on D we define Cowen–Douglas function introduced by the Cowen–Douglas operator M ϕ ∗ on Hardy space H 2 (D) , moreover, we give a necessary and sufficient condition to determine when ϕ is a Cowen–Douglas function, where ϕ ∈ H ∞ (D) and M ϕ is the associated multiplication operator on H 2 (D) . Then, we give some applications of Cowen–Douglas function on chaos, such as its application on the inverse problem of chaos for ϕ (T) , where ϕ is a Cowen–Douglas function and T is the backward shift operator on the Hilbert space L 2 (N) . [ABSTRACT FROM AUTHOR]

We show that systems with some specification properties are topologically or almost Borel universal, in the sense that any aperiodic subshift with lower entropy may be topologically or almost Borel embedded. This improves, with elementary tools, previous results of Quas and Soo [Ergodic universality of some topological dynamical systems. Trans. Amer. Math. Soc.368 (2016), 4137–4170]. [ABSTRACT FROM AUTHOR]

We prove a version of the Lefschetz fixed point theorem for multivalued maps F : X ⊸ X in which X is a finite T 0 space. [ABSTRACT FROM AUTHOR]