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

We study Laplacians on graphs and networks via regular Dirichlet forms. We give a sufficient geometric condition for essential selfadjointness and explicitly determine the generators of the associated semigroups on all ℓ p , 1 ≤ p < ∞, in this case. We characterize stochastic completeness thereby generalizing all earlier corresponding results for graph Laplacians. Finally, we study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph.

We consider discrete one-dimensional Schrödinger operators with Sturmian potentials. For a fullmeasure set of rotation numbers including the Fibonacci case we prove absence of eigenvalues for all elements in the hull.

It is shown how regular model sets can be characterized in terms of the regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map β and then relate the properties of β to those of the underlying dynamical system. As a by-product, we can show that regular model sets are, in a suitable sense, as close to periodic sets as possible among repetitive aperiodic sets.Downloaded

Abstract. We study the spectral properties of discrete one-dimensional Schrödinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely α-continuous spectrum, uniformly for all phases. The proofs rely on the unique decomposition property of Sturmian potentials, a mass-reproduction technique based upon a Gordon-type argument, and on the Jitomirskaya-Last extension of the Gilbert-Pearson theory of subordinacy.