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Journal of Statistical Physics

, Volume 174, Issue 3, pp 519–535 | Cite as

Diffraction of Return Time Measures

  • M. KesseböhmerEmail author
  • A. Mosbach
  • T. Samuel
  • M. Steffens
Article
  • 24 Downloads

Abstract

Letting T denote an ergodic transformation of the unit interval and letting \(f:[0,1)\rightarrow \mathbb {R}\) 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 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 \(\mu _{y}\) 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 \(\mu _{y}\) 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 \(\mu _{y}\) is independent of \(y.\) Finally, for a converging sequence of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.

Keywords

Aperiodic order Autocorrelation Diffraction Transformations of the unit interval Rigid rotations 

Mathematics Subject Classification

43A25 52C23 37E05 37A25 37A45 

Notes

Acknowledgements

Part of this work was completed while the authors were visiting the Mittag-Leffler Institute as part of the Research Program Fractal Geometry and Dynamics. We are extremely grateful to the organisers and staff for their very kind hospitality, financial support and stimulating atmosphere. The authors also wish to thank M. Baake and N. Strungaru for many interesting and insightful discussions. Finally, the authors are grateful to MINTernational for providing financial support for research visits between Universität Bremen and California Polytechnic State University.

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Copyright information

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Authors and Affiliations

  1. 1.FB 3 – Mathematik und InformatikUniversität BremenBremenGermany
  2. 2.Mathematics DepartmentCalifornia Polytechnic State UniversitySan Luis ObispoUSA
  3. 3.School of MathematicsUniversity of BirminghamBirminghamUK

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