# Diffraction of return time measures

Research output: Contribution to journal › Article › peer-review

## Standard

**Diffraction of return time measures.** / Kesseböhmer, Marc; Mosbach, Arne; Samuel, Tony; Steffens, Malte.

Research output: Contribution to journal › Article › peer-review

## Harvard

*Journal of Statistical Physics*, vol. 174, no. 3, pp. 519–535. https://doi.org/10.1007/s10955-018-2196-5

## APA

*Journal of Statistical Physics*,

*174*(3), 519–535. https://doi.org/10.1007/s10955-018-2196-5

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## Author

## Bibtex

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## RIS

TY - JOUR

T1 - Diffraction of return time measures

AU - Kesseböhmer, Marc

AU - Mosbach, Arne

AU - Samuel, Tony

AU - Steffens, Malte

PY - 2019/2

Y1 - 2019/2

N2 - 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 μy for a reference point y∈[0,1) as the weighted Dirac comb with support in ℤ and weights f∘Tz(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 μy 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 μy 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 μy is independent of y. Finally, for a converging sequence of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.

AB - 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 μy for a reference point y∈[0,1) as the weighted Dirac comb with support in ℤ and weights f∘Tz(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 μy 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 μy 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 μy is independent of y. Finally, for a converging sequence of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.

KW - Aperiodic order

KW - Autocorrelation

KW - Diffraction

KW - Interval maps

KW - Rigid rotations

KW - Transformations of the unit interval

UR - https://arxiv.org/abs/1801.07608

UR - http://www.scopus.com/inward/record.url?scp=85057576532&partnerID=8YFLogxK

U2 - 10.1007/s10955-018-2196-5

DO - 10.1007/s10955-018-2196-5

M3 - Article

VL - 174

SP - 519

EP - 535

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -