@article{9faefd2c6cb643e0ad2ff0a85c751d2e,
title = "Self Affine Delone Sets and Deviation Phenomena",
abstract = "We study the growth of norms of ergodic integrals for the translation action on spaces coming from expansive, self-affine Delone sets. The linear map giving the self-affinity induces a renormalization map on the pattern space and we show that the rate of growth of ergodic integrals is controlled by the induced action of the renormalizing map on the cohomology of the pattern space up to boundary errors. We explore the consequences for the diffraction of such Delone sets, and explore in detail what the picture is for substitution tilings as well as for cut and project sets which are self-affine. We also explicitly compute some examples.",
author = "Scott Schmieding and Rodrigo Trevi{\~n}o",
note = "Funding Information: Acknowledgements. R.T. wishes to thank J. Kellendonk for patiently explaining cohomology of pattern spaces, G. Forni for many insightful discussions about deviations of ergodic averages for translation actions, and B. Weiss for many useful conversations about tilings and Delone sets as well as many useful comments about an early draft of this paper which greatly improved it. R.T. was partially supported by the NSF under Award No. DMS-1204008, BSF Grant 2010428, and ERC Starting Grant DLGAPS 279893. A significant part of this was written while R.T. was visiting IMPA during the Programa de Pos-Doutorado de Verao 2015, and is grateful for the hospitality. This research was also supported in part by the National Science Foundation grant “RTG: Analysis on manifolds” at Northwestern University. We would also like to thank the participants of Arbeitsgemeinschaft: Mathematical Quasicrystals held at Mathematisches Forschungsinstitut Oberwolfach in October 2015 for many useful conversations. We are also very grateful to Lorenzo Sadun for pointing out a mistake in an early version of this paper. Finally, we are grateful to an anonymous referee for many helpful suggestions that improved the paper. Publisher Copyright: {\textcopyright} 2017, Springer-Verlag GmbH Germany.",
year = "2018",
month = feb,
day = "1",
doi = "10.1007/s00220-017-3011-x",
language = "English (US)",
volume = "357",
pages = "1071--1112",
journal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer New York",
number = "3",
}