Certain topological dynamical systems that arise from actions of $\sigma$-compact locally compact Abelian groups on compact spaces of translation bounded measures are considered. Such a measure dynamical system is shown to have a pure point dynamical spectrum if and only if its diffraction spectrum is pure point.

We introduce the notion of a space with measured walls, generalizing the concept of a space with walls due to Haglund and Paulin (Simplicité de groupes d'automorphismes d'espaces à courbure négative. Geom. Topol. Monograph1 (1998), 181–248). We observe that if a locally compact group G acts properly on a space with measured walls, then G has the Haagerup property. We conjecture that the converse holds and we prove this conjecture for the following classes of groups: discrete groups with the Haagerup property, closed subgroups of SO(n, 1), groups acting properly on real trees, SL2(K) where K is a global field and amenable groups.

We study C1 invariant, two-dimensional tori of constant energy without singularities of the Euler–Lagrange flow of a convex, superlinear Lagrangian defined in the tangent space of a closed surface. We show that in regular energy levels, such tori are graphs of the canonical projection if and only if they are minimizing. This result was known for C3 invariant, Lagrangian tori of Finsler Lagrangians, which implied the statement for C3 invariant Lagrangian tori of unimodal Lagrangians with a high-energy level. We also show that graphs have energy $E \geq c_0(L)$, where c0(L) is Mañé's strict critical value; and that the presence of Reeb components in such tori implies that the energy is c0(L). Moreover, if we assume that the Lagrangian is unimodal, we show that Lagrangian invariant tori with no singularities have no Reeb components, regardless of the energy level.

We classify the $C^\infty$ volume-preserving uniformly quasiconformal Anosov flows, such that $E^+\oplus E^-$ is $C^\infty$ and the dimensions of E+ and E- are at least two. Then we deduce a classification of volume-preserving uniformly quasiconformal Anosov flows with smooth distributions.

If $\mathbb{M}$ is a monoid and $\mathcal{A}$ is an Abelian group, then $\mathcal{A}^{\mathbb{M}}$ is a compact Abelian group; a linear cellular automaton (LCA) is a continuous endomorphism $\mathfrak{F}:\mathcal{A}^{\mathbb{M}}\longrightarrow \mathcal{A}^{\mathbb{M}}$ that commutes with all shift maps. If $\mathfrak{F}$ is diffusive, and $\mu$ is a harmonically mixing (HM) probability measure on $\mathcal{A}^{\mathbb{M}}$, then the sequence $\{\mathfrak{F}^N \mu\}_{N=1}^{\infty}$ weak* converges to the Haar measure on $\mathcal{A}^{\mathbb{M}}$ in density. Fully supported Markov measures on $\mathcal{A}^{\mathbb{Z}}$ are HM and non-trivial LCA on $\mathcal{A}^{(\mathbb{Z}^D)}$ are diffusive when $\mathcal{A}=\mathbb{Z}_{/p}$ is a prime cyclic group.

In the present work, we provide sufficient conditions for diffusion of LCA on $\mathcal{A}^{(\mathbb{Z}^D)}$ when $\mathcal{A}=\mathbb{Z}_{/n}$ is any cyclic group or when $\mathcal{A}=(\mathbb{Z}_{/p^r})^J$ (p prime). We also show that any fully supported Markov random field on $\mathcal{A}^{(\mathbb{Z}^D)}$ is HM (where $\mathcal{A}$ is any Abelian group).

Consider a closed, smooth manifold M of non-positive curvature. Write $p:\textit{UM}\rightarrow M$ for the unit tangent bundle over M and let ${\mathcal R}_>$ denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow $\phi$ on UM. We will define the structured dimension s-dim $\mathcal{R}_>$ which, essentially, is the dimension of the set $p(\mathcal{R}_>)$ of base points of $\mathcal{R}_>$.

The main result of this paper holds for manifolds with s-dim $\mathcal{R}_><\dim M/2$: for every $\epsilon>0$, there is an $\epsilon$-dense, flow invariant, closed subset $\Xi_\epsilon\subset \textit{UM}\backslash{\mathcal{R}}_>$ such that $p(\Xi_\epsilon)=M$.