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In this paper, we develop the theory of \mathbb {Z}_p-index which has been introduced by Tsukamoto, Tsutaya and Yoshinaga [ G-index, Topological Dynamics and Marker Property , arXiv: 2012.15372 , 2020]. As an application, we show that given any positive number, there exists a dynamical system with mean dimension equal to such number such that it does not have the marker property. [ABSTRACT FROM AUTHOR]

Katznelson's Question is a long-standing open question concerning recurrence in topological dynamics with strong historical and mathematical ties to open problems in combinatorics and harmonic analysis. In this article, we give a positive answer to Katznelson's Question for certain towers of skew-product extensions of equicontinuous systems, including systems of the form (x,t) \mapsto (x + \alpha, t + h(x)). We describe which frequencies must be controlled for in order to ensure recurrence in such systems, and we derive combinatorial corollaries concerning the difference sets of syndetic subsets of the natural numbers. [ABSTRACT FROM AUTHOR]

We prove that the minimal chain recurrence classes of a holomorphic endomorphism of \mathbb {P}^k have finitely many connected components. We also obtain results on arbitrary classes. These strong constraints on the topological dynamics in the phase space are all deduced from the associated action on a space of currents. [ABSTRACT FROM AUTHOR]

We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. It is proved that transitive countably expansive homeomorphisms satisfying the shadowing property are expansive in the set of transitive points. This is in contrast with pseudo-Anosov diffeomorphisms of the two-dimensional sphere that are transitive, cw-expansive, satisfy the shadowing property but the dynamical ball in each transitive point contains a Cantor subset. We exhibit examples of countably expansive homeomorphisms that are not finite expansive, satisfy the shadowing property and admits an infinite number of chain-recurrent classes. We further explore the relation between countable and entropy expansivity and prove that for surface homeomorphisms f\colon S\to S satisfying the shadowing property and \Omega (f)=S, both countably expansive and entropy cw-expansive are equivalent to being topologically conjugate to an Anosov diffeomorphism. [ABSTRACT FROM AUTHOR]

We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide classes of examples of theories with [pro]finite or trivial Ellis groups. We also find several concrete examples illustrating the lack of implications between some fundamental properties. In the appendix, we give a full computation of the Ellis group of the theory of the random hypergraph with one binary and one 4-ary relation. This example shows that the assumption of NIP in the version of Newelski's conjecture for amenable theories (proved by Krupiński, Newelski, and Simon [J. Math. Log. 19 (2019), no. 2, 1950012, p. 55]) cannot be dropped. [ABSTRACT FROM AUTHOR]

A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this property, and this class has recently attracted much attention in complex dynamics. In the presence of escaping critical values, these curves break or split at critical points. In this paper, we develop combinatorial tools that allow us to provide a complete description of the escaping set of any criniferous function without asymptotic values on its Julia set. In particular, our description precisely reflects the splitting phenomenon. This combinatorial structure provides the foundation for further study of this class of functions. For example, we use these results in another paper to give the first full description of the topological dynamics of a class of transcendental entire maps with unbounded postsingular set. [ABSTRACT FROM AUTHOR]

The paper is mainly concerned with the lifting of locally almost periodicity of minimal semiflows. Based on this work, we use a different method from Sacker and Sell to get the property of lifting of equicontinuity for semiflows. Moreover, in the case of a group action, we give answers to questions about lifting of almost automorphy for minimal flows posed by Sell, Shen, and Yi [ Topological dynamics and differential equations , Amer. Math. Soc., Providence, RI, 1998, pp. 279-298]. [ABSTRACT FROM AUTHOR]

We explain how dynamical systems with generating partitions are symbolically expansive, namely symbolic counterparts of the expansive ones. Similar ideas allow the notions of symbolic equicontinuity, symbolic distality, symbolic N-expansivity, and symbolic shadowing property. We analyze dynamical systems with these properties in the circle. Indeed, we show that every symbolically N-expansive circle homeomorphism has finitely many periodic points. Moreover, if there are no wandering points, then the situation will depend on the rotation number. In the rational case the homeomorphism is symbolically equicontinuous with the symbolic shadowing property and, in the irrational case, the homeomorphism is symbolically expansive, symbolically distal, but not symbolically equicontinuous. We will also introduce a symbolic entropy and study its properties. [ABSTRACT FROM AUTHOR]

We classify the ergodic invariant random subgroups of blockdiagonal limits of symmetric groups in the cases when the groups are simple and the associated dimension groups have finite-dimensional state spaces. These block-diagonal limits arise as the transformation groups (full groups) of Bratteli diagrams that preserve the cofinality of infinite paths in the diagram. Given a simple full group G admitting only a finite number of ergodic measures on the path-space Xn of the associated Bratteli diagram, we prove that every non-Dirac ergodic invariant random subgroup of G arises as the stabilizer distribution of the diagonal action on Xn for some n ≥ 1. As a corollary, we establish that every group character χ of G has the form χ(g) = Prob(g ∈ K), where K is a conjugation-invariant random subgroup of G. [ABSTRACT FROM AUTHOR]

We study extensions of the asymptotic density to a finitely additive measure defined on all subsets of natural numbers. Such measures are called density measures. We consider a class of density measures constructed from free ultrafilters on natural numbers and investigate absolute continuity and singularity for those density measures. In particular, for any pair of such density measures we prove necessary and sufficient conditions that one is absolutely continuous with respect to the other and that they are singular. Also we prove similar results for weak absolute continuity and strong singularity. These results are formulated in terms of topological dynamics. [ABSTRACT FROM AUTHOR]

This paper forms the second part of a series of three papers by the authors concerning the structure of nilspaces of Antolín Camarena and Szegedy. A nilspace is a compact space X together with closed collections of cubes Cn(X) ⊆ X2n, n = 1, 2, … satisfying some natural axioms. From these axioms it follows that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group. Our main result is a new proof of a result due to Antolín Camarena and Szegedy [Nilspaces, nilmanifolds and their morphisms, arXiv:1009.3825v3 (2012)] stating that if each of these groups is a torus, then X is isomorphic (in a strong sense) to a nilmanifold G/Г. We also extend the theorem to a setting where the nilspace arises from a dynamical system (X, T). These theorems are a key stepping stone towards the general structure theorem in [The structure theory of nilspaces III: Inverse limit representations and topological dynamics, arXiv:1605.08950v1 [math.DS] (2016)] (which again closely resembles the main theorem of Antolín Camarena and Szegedy). The main technical tool, enabling us to deduce algebraic information from topological data, consists of existence and uniqueness results for solutions of certain natural functional equations, again modelled on the theory in Antolín Camarena and Szegedy's paper. [ABSTRACT FROM AUTHOR]

We provide a new proof of a recent theorem of Ben Yaacov, Melleray, and Tsankov. If G is a Polish group and X is a minimal, metrizable G-flow with all orbits meager, then the universal minimal flow M(G) is nonmetrizable. In particular, we show that given X as above, the universal highly proximal extension of X is nonmetrizable. [ABSTRACT FROM AUTHOR]

In this paper we extend the concept of topological stability from homeomorphisms to group actions on compact metric spaces and prove that if an action of a finitely generated group is expansive and has the pseudoorbit tracing property, then it is topologicaly stable. This represents a group action version of P. Walter's stability theorem [Lecture Notes in Math., vol. 668, Springer, 1978, pp. 231-244]. Moreover we give a class of group actions with topological stability or pseudo-orbit tracing property. In particular, we establish a characterization of subshifts of finite type over finitely generated groups in terms of the pseudo-orbit tracing property. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group, and denote by WAP(M(G)) and AP(M(G)) the spaces of weakly almost periodic, respectively, almost periodic functionals on the measure algebra M(G). Problem 3 in [H.G. Dales, A.T.- M. Lau, D. Strauss, Second duals of measure algebras, Dissertationes Math. (Rozprawy Mat.) 481 (2012), 1-121] asks if WAP(M(G)) and AP(M(G)) admit topological invariant means, and if yes, whether they are unique. The questions regarding existence had already been raised in [M. Daws, Characterising weakly almost periodic functionals on the measure algebra, Studia Math. 204 (2011), no. 3, 213-234]. We answer all these problems in the affirmative. [ABSTRACT FROM AUTHOR]

We prove that if a topological dynamical system is mean sensitive and contains a mean proximal pair consisting of a transitive point and a periodic point, then it is mean Li-Yorke chaotic (DC2 chaotic). On the other hand we show that a system is mean proximal if and only if it is uniquely ergodic and the unique measure is supported on one point. [ABSTRACT FROM AUTHOR]

Consider, for any n ∊ N, the set Posn of all n-periodic tree patterns with positive topological entropy and the set Irrn ⊈ Posn of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families Posn and Irrn. Let λn be the unique real root of the polynomial xn - 2x - 1 in (1,+∞). We explicitly construct an irreducible n-periodic tree pattern Qn whose entropy is log(#955;n). For n = mk, where m is a prime, we prove that this entropy is minimum in the set Posn. Since the pattern Qn is irreducible, Qn also minimizes the entropy in the family Irrn. [ABSTRACT FROM AUTHOR]

For G a closed subgroup of such∞, we provide a precise combinatorial characterization of when the universal minimal flow M(G) is metrizable. In particular, each such instance fits into the framework of metrizable flows developed by Kechris, Pestov, and Todorˇcevi'c and by Nguyen Van Th'e; as a consequence, each G with metrizable universal minimal flow has the generic point property, i.e. every minimal G-flow has a point whose orbit is comeager. This solves the Generic Point Problem raised by Angel, Kechris, and Lyons for closed subgroups of S∞. [ABSTRACT FROM AUTHOR]

Let (X, T) be a topological dynamical system. Denote by h(T,K) and hB(T,K) the covering entropy and dimensional entropy of K ⊆ X, respectively. (X, T) is called D-lowerable (resp. lowerable) if for each 0 ⩽ h ⩽ h(T,X) there is a subset (resp. closed subset) Kh with hB(T,Kh) = h (resp. h(T,Kh) = h) and is called D-hereditarily lowerable (resp. hereditarily lowerable) if each Souslin subset (resp. closed subset) is D-lowerable (resp. lowerable). In this paper it is proved that each topological dynamical system is not only lowerable but also D-lowerable, and each asymptotically h-expansive system is D-hereditarily lowerable. A minimal system which is lowerable and not hereditarily lowerable is demonstrated. [ABSTRACT FROM AUTHOR]

According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group &n are all conjugate. In this paper, we study possible generalizations of this property to other types of Coxeter groups. We show that Schützenberger's result is a special case of a general result on "smooth" two-sided cells. Furthermore, we consider Kottwitz's conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types which, combined with further recent work, settles this conjecture in general. [ABSTRACT FROM AUTHOR]

A partially hyperbolic diffeomorphism f is structurally quasistable if for any diffeomorphism g C¹-close to f, there is a homeomorphism p of M such that π◦ g and f ◦π differ only by a motion t along center directions. f is topologically quasi-stable if for any homeomorphism g C0-close to f, the above holds for a continuous map π instead of a homeomorphism. We show that any partially hyperbolic diffeomorphism f is topologically quasi-stable, and if f has C¹ center foliation Wfc, then f is structurally quasi-stable. As applications we obtain continuity of topological entropy for certain partially hyperbolic diffeomorphisms with one or two dimensional center foliation. [ABSTRACT FROM AUTHOR]

We investigate some product structures in R. Thompson's group V, primarily by studying the topological dynamics associated with V's action on the Cantor set .... We draw attention to the class D (V,...) of groups which have embeddings as demonstrative subgroups of V whose class can be used to assist in forming various products. Note that D (V,...) contains all finite groups, the free group on two generators, and Q/Z, and is closed under passing to subgroups and under taking direct products of any member by any finite member. If G = V and H ∈ D (V,...), then G ... H embeds into V . Finally, if G, H ∈ D(V,...), then G * H embeds in V. Using a dynamical approach, we also show the perhaps surprising result that Z2 * Z does not embed in V, even though V has many embedded copies of Z2 and has many embedded copies of free products of various pairs of its subgroups. [ABSTRACT FROM AUTHOR]

We prove that a homogeneous matchbox manifold is homeomorphic to a McCord solenoid, thereby proving a strong version of a conjecture of Fokkink and Oversteegen, which is a general form of a conjecture of Bing. A key step in the proof shows that if the foliation of a matchbox manifold has equicontinuous dynamics, then it is minimal. Moreover, we then show that a matchbox manifold with equicontinuous dynamics is homeomorphic to a weak solenoid. A result of Effros is used to conclude that a homogeneous matchbox manifold has equicontinuous dynamics, and the main theorem is a consequence. The proofs of these results combine techniques from the theory of foliations and pseudogroups, along with methods from topological dynamics and coding theory for pseudogroup actions. These techniques and results provide a framework for the study of matchbox manifolds in general, and exceptional minimal sets of smooth foliations. [ABSTRACT FROM AUTHOR]

Let f : ℂ → ℂ be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set F(f) and the postsingular set P(f) is compact and the intersection of the Julia set ℑ(f) and P(f) is finite. Assume that no asymptotic value of f belongs to ℑ(f) and that the local degree of f at all points in ℑ(f) is bounded by some finite constant. We prove that there is a hyperbolic map g ∈ {z ↦ f(λz) : λ ∈ ℂ} with connected Fatou set such that f and g are semiconjugate on their Julia sets. Furthermore, we show that this semiconjugacy is a conjugacy when restricted to the escaping set I(g) of g. In the case where f can be written as a finite composition of maps of finite order, our theorem, together with recent results on Julia sets of hyperbolic maps, implies that ℑ(f) is a pinched Cantor bouquet, consisting of dynamic rays and their endpoints. Our result also seems to give the first complete description of topological dynamics of an entire transcendental map whose Julia set is the whole complex plane. [ABSTRACT FROM AUTHOR]

The unit sphere $ S$ $ \mathbb{C}^n$ $ \mathcal{L}$ $ \mathcal{L}$, that is, half the topological dimension of $ S$ [ABSTRACT FROM AUTHOR]

We introduce the notion of topological entropy dimension to measure the complexity of entropy zero systems. It measures the superpolynomial, but subexponential, growth rate of orbits. We also introduce the dimension set, $ \mathcal{D}(X,T)\subset [0,1]$-mixing property in measurable dynamics and to the uniformly positive entropy in topological dynamics for positive entropy systems. Using the entropy dimension, we are able to discuss the disjointness between the entropy zero systems. Properties of entropy generating sequences and their dimensions have been investigated. [ABSTRACT FROM AUTHOR]

According to an analogy to quasi-Fuchsian groups, we investigate the topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic $3$-laminations associated with hyperbolic and parabolic quadratic maps. par We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same $3$-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to-parabolic degenerations (and parabolic-to-hyperbolic bifurcations) of quadratic maps. For example, the differences between the structures of the quotient $3$-laminations of Douady's rabbit, the Cauliflower, and $z mapsto z^2$ are described. par The descriptions employ a new method of textit {tessellation} inside the filled Julia set introduced in Part I [textit {Ergodic Theory Dynam. Systems} {bfseries 29} (2009), no. 2] that works like external rays outside the Julia set. [ABSTRACT FROM AUTHOR]

It is proved that the infimum of the topological entropy of continuous topologically exact interval (circle) maps is strictly smaller than the infimum of the topological entropy of continuous interval (circle) maps, which are topologically mixing, but not exact. Interpreting this result in terms of popular notions of chaos, one may say that on the interval (circle) lower entropy implies stronger Devaney chaos. Moreover, the infimum of the entropy of mixing circle maps is computed. These theorems may be considered as a completion of some results of Alsedà, Kolyada, Llibre, and Snoha (1999). [ABSTRACT FROM AUTHOR]

We construct two equivalent smooth flows, one of which has positive topological entropy and the other has zero topological entropy. This provides a negative answer to a problem posed by Ohno. [ABSTRACT FROM AUTHOR]

We study topological dynamics on {\em unshielded} planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial $f$ with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure $\mu$ this easily implies that one of the following holds: 1. for $\mu$-a.e. $x\in J(f)$, $\omega(x)=J(f)$; 2. for $\mu$-a.e. $x\in J(f)$, $\omega(x)=\omega(c(x))$ for a critical point $c(x)$ depending on $x$. [ABSTRACT FROM AUTHOR]

In these notes we explore the fine structure of recurrence for semigroup actions, using the algebraic structure of compactifications of the acting semigroup. [ABSTRACT FROM AUTHOR]

$IP^*$ sets and central sets are subsets of $\mathbb N$ which arise out of applications of topological dynamics to number theory and are known to have rich combinatorial structure. Spectra of numbers are often studied sets of the form $\{[n\alpha+\gamma]\colon n\in\mathbb N\}$. Iterated spectra are similarly defined with $n$ coming from another spectrum. Using elementary, dynamical, and algebraic approaches we show that iterated spectra have significantly richer combinatorial structure than was previously known. For example we show that if $\alpha>0$ and $0<\gamma<1$, then $\{[n\alpha+\gamma]\colon n\in\mathbb N\}$ is an $IP^*$ set and consequently contains an infinite sequence together with all finite sums and products of terms from that sequence without repetition. [ABSTRACT FROM AUTHOR]

We give a new short proof for the uniqueness of the universal minimal space. The proof holds for the uniqueness of the universal object in every collection of topological dynamical systems closed under taking projective limits and possessing universal objects. [ABSTRACT FROM AUTHOR]