We show that for some simple classical chaotic dynamical systems the set of Li-Yorke pairs has full Hausdorff dimension on invariant sets.

We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that in degree one this map is injective and has torsion free cokernel. We show by example, however, that, in degree one, the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology.

Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on $\Phi \subset 2^{2^X}$, the space of maximal chains in $2^X$, equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on $U_{Homeo\left(X\right)}$, the universal minimal space of Homeo(X), is not transitive (improving...

We prove that on a metrizable, compact, zero-dimensional space every ${\mathbb{Z}}^d$-action with no periodic points is measurably isomorphic to a minimal ${\mathbb{Z}}^d$-action with the same, i.e. affinely homeomorphic, simplex of measures.

We show that there are (1) nonhomogeneous metric continua that admit minimal noninvertible maps but have the fixed point property for homeomorphisms, and (2) nonhomogeneous metric continua that admit both minimal noninvertible maps and minimal homeomorphisms. The former continua are constructed as quotient spaces of the torus or as subsets of the torus, the latter are constructed as subsets of the torus.

We construct a continuous non-invertible minimal transformation of an arbitrary solenoid. Since solenoids, as all other compact monothetic groups, also admit minimal homeomorphisms, our result allows one to classify solenoids among continua admitting both invertible and non-invertible continuous minimal maps.

As was known to H. Poincaré, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms, that is, torus homeomorphisms isotopic to the identity for which the rotation...

In this paper we investigate numerous constructions of minimal systems from the point of view of $({ℱ}_1,{ℱ}_2)$-chaos (but most of our results concern the particular cases of distributional chaos of type $1$ and $2$). We consider standard classes of systems, such as Toeplitz flows, Grillenberger $K$-systems or Blanchard-Kwiatkowski extensions of the Chacón flow, proving that all of them are DC2. An example of DC1 minimal system with positive topological entropy is also introduced. The above mentioned results answer...

In 2005, the paper [KPT05] by Kechris, Pestov and Todorcevic provided a powerful tool to compute an invariant of topological groups known as the universal minimal flow. This immediately led to an explicit representation of this invariant in many concrete cases. However, in some particular situations, the framework of [KPT05] does not allow one to perform the computation directly, but only after a slight modification of the original argument. The purpose of the present paper is to supplement [KPT05]...

We examine multiple disjointness of minimal flows, that is, we find conditions under which the product of a collection of minimal flows is itself minimal. Our main theorem states that, for a collection ${X_i}_{i\in I}$ of minimal flows with a common phase group, assuming each flow satisfies certain structural and algebraic conditions, the product ${\prod }_{i\in I}{X}_i$ is minimal if and only if ${\prod }_{i\in I}{X}_i^{eq}$ is minimal, where $X_i^{eq}$ is the maximal equicontinuous factor of $X_i$. Most importantly, this result holds when each $X_i$ is distal. When the phase...

For a topological group G we introduce the algebra SUC(G) of strongly uniformly continuous functions. We show that SUC(G) contains the algebra WAP(G) of weakly almost periodic functions as well as the algebras LE(G) and Asp(G) of locally equicontinuous and Asplund functions respectively. For the Polish groups of order preserving homeomorphisms of the unit interval and of isometries of the Urysohn space of diameter 1, we show that SUC(G) is trivial. We introduce the notion of fixed point on a class...

En topologie dynamique, une famille classique de systèmes est celle formée par les rotations minimales. La classe des nilsystèmes et de leurs limites projectives en est une extension naturelle. L’étude de ces systèmes est ancienne mais connaît actuellement un renouveau à cause de ses applications, à la fois à la théorie ergodique et en théorie additive des nombres. Les rotations minimales sont caractérisées par le fait que la relation de proximalité régionale est l’égalité. Nous introduisons une...

We consider continuous extensions of minimal rotations on a locally connected compact group X by cocycles taking values in locally compact Lie groups and prove regularity (i.e. the existence of orbit closures which project onto the whole basis X) in certain special situations beyond the nilpotent case. We further discuss an open question on cocycles acting on homogeneous spaces which seems to be the missing key for a general regularity theorem.

For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f(A) and $f^{-1}\left(A\right)$ share with A those topological properties which describe how large a set is. Using these results...

Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett-Krieger models of such rotations. In this paper we show that it does, as long as the model is at the same a topological extension, via the same map that establishes the isomorphism, of an equicontinuous model. In particular, we recover (after [AKL]) that regular Toeplitz systems satisfy Sarnak's conjecture, and, as another consequence, so do...

We study relations between the almost specification property, the asymptotic average shadowing property and the average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples showing that compactness is a necessary condition for these implications to hold. As a consequence, we also obtain a proof that limit shadowing in...

We continue the study of topological properties of the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the uniform topology τ, which was started by Bezuglyi, Dooley, Kwiatkowski and Medynets. We prove that the set of periodic homeomorphisms is τ-dense in Homeo(X) and deduce from this result that the topological group (Homeo(X),τ) has the Rokhlin property, i.e., there exists a homeomorphism whose conjugacy class is τ-dense in Homeo(X). We also show that for any homeomorphism...

In the paper the fundamental properties of discrete dynamical systems generated by an $\alpha$-condensing mapping ($\alpha$ is the Kuratowski measure of noncompactness) are studied. The results extend and deepen those obtained by M. A. Krasnosel’skij and A. V. Lusnikov in [21]. They are also applied to study a mathematical model for spreading of an infectious disease investigated by P. Takac in [35], [36].

We discuss some results about derivations and crossed homomorphisms arising in the context of locally compact groups and their group algebras, in particular, L¹(G), the von Neumann algebra VN(G) and actions of G on related algebras. We answer a question of Dales, Ghahramani, Grønbæk, showing that L¹(G) is always permanently weakly amenable. Then we show that for some classes of groups (e.g. IN-groups) the homology of L¹(G) with coefficients in VN(G) is trivial. But this is no longer true, in general,...

We consider a class of symbolic systems over a finite alphabet which are minimal almost one-to-one extensions of rotations of compact metric monothetic groups and provide computations of their enveloping semigroups that highlight their algebraic structure. We describe the set of idempotents of these semigroups and introduce a classification that can help distinguish between certain such systems having zero topological entropy.