RESEARCH SCHOOL - ECOLE DE RECHERCHE
Tiling Dynamical System (1720)
Pavages et systèmes dynamiques
Dates:20-24 November 2017 at 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.
SCIENTIFIC COMMITTEE
Boris Adamczewski (Aix-Marseille Université)
Valérie Berthé (Université Paris Diderot)
Anne Siegel (IRISA CNRS Rennes)
Boris Solomyak (University of Bar Ilan)
ORGANIZING COMMITTEE
Shigeki Akiyama (University of Tsukuba & Aix-Marseille Université)
Pierre Arnoux (Aix-Marseille Université) 
MAIN SPEAKERS
Introduction to hierarchical tiling dynamical systems
Lecture 1 -  Lecture2 - Lecture3 - VIDEO

Undecidability of the Domino Problem 
Lecture 1 - Lecture 2 - Lecture 3  - VIDEO

​Operators, Algebras and their Invariants for Aperiodic Tilings

From combinatorial games to shape-symmetric morphisms​
Lectures 1-3 - Follow-up - VIDEO

Delone sets and Tilings​   
Lecture - VIDEO

S-adic sequences A bridge between dynamics, arithmetic, and geometry​​
Lecture 1 - Lecture 2 - Lecture 3 -  VIDEO

OTHER PRESENTERS
  • Paulina Cecchi (Université Paris Diderot)
​Invariant measures for actions of congruent monotilable amenable groups   (pdf)
  • Jungwon Lee (UNIST South Korea)
Distribution of modular symbols: a dynamical approach
  • Ivan Mitrofanov (ENS Paris)
Algorithmical properties of transducer groups and tilings
  • J.M. Rodriguez Caballero (UQAM)
Balanced parentheses and the E-polynomials of the Hilbert scheme of n points on a torus   (pdf)
  • Filipp Rukhovich (Moscow Insitute of Physics)
Outer billiards outside regular polygons: sets of full measure and aperiodic points  (pdf) (poster)
  • Mao Shinoda (Keio University)
Uncountably Many Ergodic Maximizing Measures for Dense Continuous Functions   (pdf)​
  • Yotam Smilansky (Tel-Aviv University)
Kakutani’s splitting procedure for substitution partitions
  • Yaar Solomon (Ben-Gurion University)
Tiling Dynamical Systems  (pdf)
  • Yuki Takahashi (Bar-Ilan University)
Products of two Cantor sets and application to the Labyrinth model   (pdf)
  • Shuqin Zhang (Montanuniversität Leoben)
The space-filling curve of self-similar sets: two examples  (pdf)
SPONSORS