Jean-Morlet chair > shigeki akiyama - pierre arnoux (semester 2, 2017)

RESEARCH SCHOOL
Tiling Dynamical System (event 1720)
Dates: 20-24 November 2017
Place: CIRM (Marseille Luminy, France)
Description
Tiling dynamical system gives a generalization of substitutive dynamical system. It gives a nice model of quasi-crystals, recognized as another new stable state of real materials. International experts on this topic will meet PhD students interested in this developing area.
 
Basic terminology in tiling and point sets 
Tiling is a classical object. We first come back to the basic problem of its classification. Then we prepare notation of associated point sets and review basic notation of Delone set, Meyer set, Patterson set etc.
 
Spectral property of tiling dynamical systems 
To deal with tiling dynamical system, we discuss its topology, the dynamical hull, minimality and unique ergodicity. Under unique ergodicity, we may discuss its spectral property in detail (see. [4, 1]). Tiling dynamical system can be produced by a finite amount of data if we have self-affine expansion and we review results in this case.
 
Recurrence property of tilings 
When tiling is produced by cut and projection, its dynamical system shows pure discrete spectrum. In fact the converse almost holds (c.f. [3]). In such a pure discrete case, there are new developments on bounded remainder sets, and many classical Diophantine problems come into this field (see [2]). We shall discuss this connection during this course.
 
References
[1] M. Baake and R.V. Moody, Weighted dirac combs with pure point diffraction, J. Reine Angew. Math. 573 (2004), 61-94.
[2] A. Haynes, H. Koivusalo, and J. Walton, Super perfectly ordered quasicrystals and the littlewood conjecture, ArXiv:1506.05649.
[3] J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets, J. Geom. Phys. 57 (2007), no. 11, 2263-2285.
[4] B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 695-738.
 
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