Diffraction of return time measures

TL;DR

This paper studies the diffraction spectra of return time measures generated by ergodic transformations, revealing how their spectral types depend on the mixing properties of the transformation, with explicit results for rotations and mixing systems.

Contribution

It introduces a new framework for analyzing diffraction of weighted return time measures and characterizes their spectral types for different classes of transformations, including mixing and rigid rotations.

Findings

Diffraction of mixing transformations has an absolutely continuous component and a trivial atom.

Diffraction of rigid rotations is pure point and independent of the reference point y.

For irrational rotations, diffraction remains stable under different observables and converging rotation sequences.

Abstract

Letting $T$ denote an ergodic transformation of the unit interval and letting $f \colon [0,1)\to \mathbb{R}$ denote an observable, we construct the $f$-weighted return time measure $\mu_y$ for a reference point $y\in[0,1)$ as the weighted Dirac comb with support in $\mathbb{Z}$ and weights $f \circ T^z(y)$ at $z\in\mathbb{Z}$, 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 which emerges from the Fourier transform of its autocorrelation and analyse it for the dependence on the underlying transformation. For certain rapidly mixing transformations and observables of bounded variation, we show that the diffraction of $\mu_{y}$ 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 the family of rigid rotations $T_{\alpha} \colon x \to x + \alpha \bmod{1}$ with rotation number $\alpha \in \mathbb{R}^+$. In contrast to when $T$ is mixing, we observe that the diffraction of $\mu_{y}$ is pure point, almost surely with respect to $y$. Moreover, if $\alpha$ is irrational and the observable $f$ is Riemann integrable, then the diffraction of $\mu_{y}$ is independent of $y$. Finally, for a converging sequence $(\alpha_{i})_{i \in \mathbb{N}}$ of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.