Model sets are always Meyer sets but the converse is generally not true. In this work we show that for a repetitive Meyer multiple set of $\mathbb{R}^{d}$ with associated dynamical system $(\mathbb{X},\mathbb{R}^{d})$ , the property of being a model multiple set is equivalent to $(\mathbb{X},\mathbb{R}^{d})$ being almost automorphic. We deduce this by showing that a repetitive Meyer multiple set can always be embedded into a repetitive model multiple set having a smaller group of topological eigenvalues.

We prove a rigidity theorem for dominated Hölder cocycles with values on diffeomorphism groups of a compact manifold over hyperbolic homeomorphisms. More precisely, we show that if two such cocycles have equal periodic data, then they are cohomologous.

We consider actions of locally compact groups $G$ on certain CAT(0) spaces $X$ by isometries. The CAT(0) spaces we consider have finite dimension at large scale. In case $B$ is a $G$-boundary, that is a measurable $G$-space with some amenability and ergodicity properties, we prove the existence of equivariant maps from $B$ to the visual boundary $\partial X$.

We show that the twisted group $C^{\ast }$ -algebra associated with a discrete FC-hypercentral group is simple (respectively, has a unique tracial state) if and only if Kleppner’s condition is satisfied. This generalizes a result of Packer for countable nilpotent groups. We also consider a larger class of groups, for which we can show that the corresponding reduced twisted group $C^{\ast }$ -algebras have a unique tracial state if and only if Kleppner’s condition holds.

We define a generic algorithmic framework to prove a pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi–Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamical properties of the Rauzy fractals, of the underlying symbolic dynamical systems, as well as some number-theoretical properties of the associated Pisot numbers.

In this article, Conley’s connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex $(C,{\rm\Delta})$ are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence $(E^{r},d^{r})$. The local version of this theorem relates differentials $d^{r}$ of the $r$th page $E^{r}$ to Smale’s theorem on cancellation of critical points.

Let $f$ be a germ of holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $\mathbb{C}$ (i.e. $f(0)=0,f^{\prime }(0)=e^{2{\it\pi}i{\it\alpha}},{\it\alpha}\in \mathbb{R}-\mathbb{Q}$). Perez-Marco showed the existence of a unique family of non-trivial invariant full continua containing the fixed point called Siegel compacta. When $f$ is non-linearizable (i.e. not holomorphically conjugate to the rigid rotation $R_{{\it\alpha}}(z)=e^{2{\it\pi}i{\it\alpha}}z$) the invariant compacts obtained are called hedgehogs. Perez-Marco developed techniques for the construction of examples of non-linearizable germs [Fixed points and circle maps. Acta Math. 179(2) (1997), 243–294]; these were used by the author to construct hedgehogs of Hausdorff dimension one, and adapted by Cheritat [Relatively compact Siegel disks with non-locally connected boundaries, Preprint, 2009] to construct Siegel disks with pseudo-circle boundaries. We use these techniques to construct hedgehogs of positive area and hedgehogs with inaccessible fixed points.

We show that the set of points on $C^{1}$ curves which are badly approximable by rationals in a number field form a winning set in the sense of Schmidt. As a consequence, we obtain a number field version of Schmidt’s conjecture in Diophantine approximation.

We consider a class of maps from the unit square to itself preserving a contracting foliation and inducing a one-dimensional map having an absolutely continuous invariant measure. We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error with respect to the Wasserstein distance. We present a rigorous implementation of our algorithm using interval arithmetics, and the result of the computation on a non-trivial example of a Lorenz-like two-dimensional map and its attractor, obtaining a statement on its local dimension.

To a Toeplitz flow $(X,T)$ we associate an ordered $K^{0}$-group, denoted $K^{0}(X,T)$, which is order isomorphic to the $K^{0}$-group of the associated (non-commutative) $C^{\ast }$-crossed product $C(X)\rtimes _{T}\mathbb{Z}$. However, $K^{0}(X,T)$ can be defined in purely dynamical terms, and it turns out to be a complete invariant for (strong) orbit equivalence. We characterize the $K^{0}$-groups that arise from Toeplitz flows $(X,T)$ as exactly those simple dimension groups $(G,G^{+})$ that contain a non-cyclic subgroup $H$ of rank one that intersects $G^{+}$ non-trivially. Furthermore, the Bratteli diagram realization of $(G,G^{+})$ can be chosen to have the ERS property, i.e. the incidence matrices of the Bratteli diagram have equal row sums. We also prove that for any Choquet simplex $K$ there exists an uncountable family of pairwise non-orbit equivalent Toeplitz flows $(X,T)$ such that the set of $T$-invariant probability measures $M(X,T)$ is affinely homeomorphic to $K$, where the entropy $h(T)$ may be prescribed beforehand. Furthermore, the analogous result is true if we substitute strong orbit equivalence for orbit equivalence, but in that case we can actually prescribe both the entropy and the maximal equicontinuous factor of $(X,T)$. Finally, we present some interesting concrete examples of dimension groups associated to Toeplitz flows.

We introduce multifractal pressure and dynamical multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the fine multifractal spectra of graph-directed self-conformal measures and the fine multifractal spectra of ergodic Birkhoff averages of continuous functions on graph-directed self-conformal sets.

We consider the dynamical system given by an $\text{Ad}$ -diagonalizable element $a$ of the $\mathbb{Q}_{p}$ -points $G$ of a unimodular linear algebraic group acting by translation on a finite volume quotient $X$ . Assuming that this action is exponentially mixing (e.g. if $G$ is simple) we give an effective version (in terms of $K$ -finite vectors of the regular representation) of the following statement: If ${\it\mu}$ is an $a$ -invariant probability measure with measure-theoretical entropy close to the topological entropy of $a$ , then ${\it\mu}$ is close to the unique $G$ -invariant probability measure of $X$ .

We study the Euler–Lagrange flow of a Tonelli Lagrangian on the 2-torus $\mathbb{T}^{2}$ at a fixed energy level ${\mathcal{E}}\subset T\mathbb{T}^{2}$ strictly above Mañé’s strict critical value. We prove that, if for some rational direction ${\it\zeta}\in S^{1}$ there is no invariant graph ${\mathcal{T}}\subset {\mathcal{E}}$ over $\mathbb{T}^{2}$ for the Euler–Lagrange flow with the property that all orbits on ${\mathcal{T}}$ have an asymptotic direction equal to ${\it\zeta}$, then there are chaotic dynamics in ${\mathcal{E}}$. This implies that, if the topological entropy of the Euler–Lagrange flow in ${\mathcal{E}}$ vanishes, then in ${\mathcal{E}}$ there are invariant graphs for all asymptotic directions ${\it\zeta}\in S^{1}$ and integrable-like behavior on a large scale.