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

DescriptionTiling 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 CommitteeOrganizing Committee- Shigeki Akiyama (University of Tsukuba) - Jean-Morlet Chair
- Pierre Arnoux (Aix-Marseille Université) - Local project leader
SpeakersIntroduction to hierarchical tiling dynamical systems - VIDEOUndecidability of the Domino Problem (Lecture 1) (Lecture 2) (Lecture 3) - VIDEO Operators, Algebras and their Invariants for Aperiodic TilingsFrom combinatorial games to shape-symmetric morphisms(pdf) (pdf-2) VIDEO Delone sets and Tilings (pdf) 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) |