We introduce the notions ‘virtual automorphism group’ of a minimal flow and ‘semiregular flow’ and investigate the relationship between the virtual and actual group of automorphisms.

We study homeomorphisms of a Cantor set with $k$ ($k<+\infty$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where $k\geq 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition $(X,\unicode[STIX]{x1D70E})$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least $k$ vertices at each level, and the image of the index map must consist of infinitesimals.

In this paper we study topological properties of the right action by translation of the Weyl chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing.

We prove that for any countable group $\unicode[STIX]{x1D6E4}$, there exists a free minimal continuous action $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{C}}$ on the Cantor set admitting an invariant Borel probability measure.

In L. W. Flinn’s PhD thesis published in 1972, the author conjectured that weakly expansive flows are also expansive flows. In this paper we use the horocycle flow on compact Riemann surfaces of constant negative curvature to show that Flinn’s conjecture is not true.

In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$.

We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalizes work of Julien where the almost canonical condition is assumed. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate by counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets.

We study lattice embeddings for the class of countable groups $\unicode[STIX]{x1D6E4}$ defined by the property that the largest amenable uniformly recurrent subgroup ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is continuous. When ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ comes from an extremely proximal action and the envelope of ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is coamenable in $\unicode[STIX]{x1D6E4}$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $\unicode[STIX]{x1D6E4}$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups.

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$, where $C$ is a finite group and $F$ a non-abelian free group.

Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\unicode[STIX]{x2202}(G,H)$. In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\unicode[STIX]{x1D6E5}(G,H)$, namely the simplex of measures on $\unicode[STIX]{x2202}(G,H)$. We determine the boundary $\unicode[STIX]{x2202}(G,H)$ in a number of cases, highlighting properties that might appear unexpected.

In this paper we continue the study of automorphism groups of constant-length substitution shifts and also their topological factors. We show that, up to conjugacy, all roots of the identity map are letter-exchanging maps, and all other non-trivial automorphisms arise from twisted compressions of another constant-length substitution. We characterize the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant-length substitution shift is topologically conjugate to a constant-length substitution shift via a letter-to-letter code.

We establish various new results on a problem proposed by Mahler [Some suggestions for further research. Bull. Aust. Math. Soc.29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich, and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover, we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle-third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We highlight two of them: firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets; and secondly, we find properties of the denominator structure of rational points in ‘missing-digit’ Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

In this paper, we study the set

$$\begin{eqnarray}A=\{p(n)+2^{n}d~\text{mod}~1:n\geq 1\}\subset [0,1],\end{eqnarray}$$

$p$

$d$

$a\in [0,\infty )$

$a~\text{mod}~1$

$a$

$A$