Substitution-based sequences with absolutely continuous diffraction

Exchangeable Sequences Driven by an Absolutely Continuous Random Measure

Let S be a Polish space and (Xn : n ≥ 1) an exchangeable sequence of S-valued random variables. Let αn(·) = P ( Xn+1 ∈ · | X1, . . . , Xn ) be the predictive measure and α a random probability measure on S such that αn weak −→ α a.s.. Two (related) problems are addressed. One is to give conditions for α λ a.s., where λ is a (non random) σ-finite Borel measure on S. Such conditions should concer...

On Absolutely Continuous Transformations

Non-absolutely Continuous Foliations

We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also ...

Geometric Properties of a Binary Non-pisot Inflation and Absence of Absolutely Continuous Diffraction

One of the simplest non-Pisot substitution rules is investigated in its geometric version as a tiling with intervals of natural length as prototiles. Via a detailed renormalisation analysis of the pair correlation functions, we show that the diffraction measure cannot comprise any absolutely continuous component. This implies that the diffraction, apart from a trivial Bragg peak at the origin, ...

Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

We consider the Laplacian on a rooted metric tree graph with branching number K ≥ 2 and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generali...