Diffraction of Return Time Measures

Letting T denote an ergodic transformation of the unit interval and letting f : [ 0 , 1 ) → R denote an observable, we construct the f -weighted return time measure μ y for a reference point y ∈ [ 0 , 1 ) as the weighted Dirac comb with support in Z and weights f ∘ T z ( y ) at z ∈ Z , and if T is non-invertible, then we set the weights equal to zero for all z < 0 . Given such a Dirac comb, we are interested in its diffraction spectrum and analyse it for the dependence on the underlying transformation. Under certain regularity conditions imposed on the interval map and the observable we explicitly calculate the diffraction of μ y which consists of a trivial atom and an absolutely continuous part, almost surely with respect to y . This contrasts what occurs in the setting of regular model sets arising from cut and project schemes and deterministic incommensurate structures. As a prominent example of non-mixing transformations, we consider rigid rotations. In this situation we observe that the diffraction of μ y is pure point, almost surely with respect to y and, if the rotation number is irrational and the observable is Riemann integrable, then the diffraction of μ y is independent of y . Finally, for a converging sequence of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.