Super-Ergodic operators

Generic Singular Spectrum for Ergodic Schrödinger Operators

We consider Schrödinger operators with ergodic potential Vω(n) = f (T (ω)), n ∈ Z, ω ∈ , where T : → is a nonperiodic homeomorphism. We show that for generic f ∈ C( ), the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani theory.

Generic Continuous Spectrum for Ergodic Schrödinger Operators

We consider discrete Schrödinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon’s Lemma that for a generic continuous sam...

Strictly Ergodic Subshifts and Associated Operators

We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twen...

Subordination and Super Ordination Results Involving Certain Convolution Operators

Algebraic Representations of Ergodic Actions and Super-rigidity

We revisit Margulis-Zimmer Super-Rigidity and provide some generalizations. In particular we obtain super-rigidity results for lattices in higherrank groups or product of groups, targeting at algebraic groups over arbitrary fields with absolute values. We also obtain cocycle super-rigidity results for a wide class of groups with respect to mixing actions. Our approach is based on a systematic s...