We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle $𝕋$. A set of integers is called $r$-Bohr if it is recurrent for all products of $r$ rotations on $𝕋$, and Bohr if it is recurrent for all products of rotations on $𝕋$. It is a result due to Katznelson that for each $r\ge 1$ there exist sets of integers which are $r$-Bohr but not $(r+1)$-Bohr. We present new examples of $r$-Bohr sets which are not Bohr, thanks to a construction which...

Let (G,X) be a transformation group, where X is a locally compact Hausdorff space and G is a compact group. We investigate the stable rank and the real rank of the transformation group C*-algebra C₀(X)⋊ G. Explicit formulae are given in the case where X and G are second countable and X is locally of finite G-orbit type. As a consequence, we calculate the ranks of the group C*-algebra C*(ℝⁿ ⋊ G), where G is a connected closed subgroup of SO(n) acting on ℝⁿ by rotation.

We prove that the automorphism group of the random lattice is not amenable, and we identify the universal minimal flow for the automorphism group of the random distributive lattice.

If φ is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Φ on the tiling space ${}_{\Phi }$ factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Φ-periodic asymptotic tilings corresponds to a finite set that projects onto the branch locus in a d-torus. We prove that if two such tiling spaces are homeomorphic, then the resulting branch loci are the same up to the action of certain affine maps on the torus.

We consider dynamical systems of the form $(X,f)$ where $X$ is a compact metric space and $f:X\to X$ is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract $\omega$-limit sets, answering a question by Will Brian.

It is shown that each expansive group action on a Peano continuum having a free dendrite must have a ping-pong game, and has positive geometric entropy when the acting group is finitely generated. As a corollary, it is shown that each Peano continuum having a free dendrite admits no expansive nilpotent group actions.

Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact noncompact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one-point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. We show that for the topological group G = Homeo(E) of self-homeomorphisms...

Let ℝ be the real line and let Homeo₊(ℝ) be the orientation preserving homeomorphism group of ℝ. Then a subgroup G of Homeo₊(ℝ) is called tightly transitive if there is some point x ∈ X such that the orbit Gx is dense in X and no subgroups H of G with |G:H| = ∞ have this property. In this paper, for each integer n > 1, we determine all the topological conjugation classes of tightly transitive subgroups G of Homeo₊(ℝ) which are isomorphic to ℤⁿ and have countably many nontransitive points.

We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M(G_{)}$ of the automorphism group $G$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of $M(G_{)}$. We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M(G_{)}$ admits a unique invariant...

For a continuous map f preserving orbits of an aperiodic ${ℝ}^m$-action on a compact space, its displacement function assigns to x the “time” $t\in {ℝ}^m$ it takes to move x to f(x). We show that this function is continuous if the action is minimal. In particular, f is homotopic to the identity along the orbits of the action.

A subset S of a topological dynamical system (X,f) containing at least two points is called a scrambled set if for any x,y ∈ S with x ≠ y one has $limin{f}_{n\to \infty }d(fⁿ\left(x\right),fⁿ\left(y\right))=0$ and $limsu{p}_{n\to \infty }d(fⁿ\left(x\right),fⁿ\left(y\right))>0$, d being the metric on X. The system (X,f) is called Li-Yorke chaotic if it has an uncountable scrambled set. These notions were developed in the context of interval maps, in which the existence of a two-point scrambled set implies Li-Yorke chaos and many other chaotic properties. In the present paper we address several questions about scrambled...

Let ϕ:G → Homeo₊(ℝ) be an orientation preserving action of a discrete solvable group G on ℝ. In this paper, the topological transitivity of ϕ is investigated. In particular, the relations between the dynamical complexity of G and the algebraic structure of G are considered.

We construct an example of two commuting homeomorphisms S, T of a compact metric space X such that the union of all minimal sets for S is disjoint from the union of all minimal sets for T. In other words, there are no common minimal points. This answers negatively a question posed in [C-L]. We remark that Furstenberg proved the existence of "doubly recurrent" points (see [F]). Not only are these points recurrent under both S and T, but they recur along the same sequence of powers. Our example shows...

In this article we study the structure of the set of weakly product recurrent points. Among others, we provide necessary conditions (related to topological weak mixing) which imply that the set of weakly product recurrent points is residual. Additionally, some new results about the class of systems disjoint from every minimal system are obtained.

We study transitive non-minimal ℕ-actions and ℤ-actions. We show that there are such actions whose non-transitive points are periodic and whose topological entropy is positive. It turns out that such actions can be obtained by perturbing minimal systems under some reasonable assumptions.

For measures on a Cantor space, the demand that the measure be "good" is a useful homogeneity condition. We examine the question of when a Bernoulli measure on the sequence space for an alphabet of size n is good. Complete answers are given for the n = 2 cases and the rational cases. Partial results are obtained for the general cases.