Planar dynamical systems with pure Lebesgue diffraction spectrum

Details Planar dynamical systems with pure Lebesgue diffraction spectrum Planar dynamical systems with pure Lebesgue diffraction spectrum

2010-03-08

arxiv.org/abs/1003.1536v2

Journal of Statistical Physics 140 (2010) 90-102

Mathematics

Dynamical Systems

14 pages, slightly revised version

Scientific paper

10.1007/s10955-010-9984-x

We examine the diffraction properties of lattice dynamical systems of algebraic origin. It is well-known that diverse dynamical properties occur within this class. These include different orders of mixing (or higher-order correlations), the presence or absence of measure rigidity (restrictions on the set of possible shift-invariant ergodic measures to being those of algebraic origin), and different entropy ranks (which may be viewed as the maximal spatial dimension in which the system resembles an i.i.d.\ process). Despite these differences, it is shown that the resulting diffraction spectra are essentially indistinguishable, thus raising further difficulties for the inverse problem of structure determination from diffraction spectra. Some of them may be resolved on the level of higher-order correlation functions, which we also briefly compare.

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