Mathematical diffraction theory of deterministic and stochastic structures: An informal summary

Baake, Michael and Grimm, Uwe (2011). Mathematical diffraction theory of deterministic and stochastic structures: An informal summary. In: Mathematics of Quasi-Periodic Order, 21-23 Jun 2010, RIMS, Kyoto, Japan.

URL: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/c...

Abstract

Mathematical diffraction theory is concerned with the determination of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra has improved considerably. Moreover, the phenomenon of homometry shows various unexpected new facets. This is particularly so when systems with stochastic components are taken into account.

After a brief introduction and a summary of pure point spectra, we discuss classic deterministic examples with singular or absolutely continuous spectra. In particular, we present an isospectral family of structures with continuously varying entropy. We augment this with more recent results on the diffraction of dynamical systems of algebraic origin and various further systems of stochastic nature. A systematic approach is mentioned via the theory of stochastic processes.

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