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# 2017 - Sem 2 - Akiyama - Arnoux  | enregistrements trouvés : 10

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## Post-edited  Introduction to hierarchical tiling dynamical systems: Supertile construction methods Frank, Natalie Priebe (Auteur de la Conférence) | CIRM (Editeur )

These lectures introduce the dynamical systems approach to tilings of Euclidean space, especially quasicrystalline tilings that have been constructed using a ‘supertile method’. Because tiling dynamics parallels one-dimensional symbolic dynamics, we discuss this case as well, highlighting the differences and similarities in the methods of study and the results that can be obtained.
In the first lecture we motivate the field with the discovery of quasicrystals, which led to D. Schectman’s winning the 2011 Nobel Prize in Chemistry. Then we set up the basics of tiling dynamics, describing tiling spaces, a tiling metric, and the shift or translation actions. Shift-invariant and ergodic measures are discussed, along with fundamental topological and dynamical properties.
The second lecture brings in the supertile construction methods, including symbolic substitutions, self-similar tilings, $S$-adic systems, and fusion rules. Numerous examples are given, most of which are not the “standard” examples, and we identify many commonalities and differences between these interrelated methods of construction. Then we compare and contrast dynamical results for supertile systems, highlighting those key insights that can be adapted to all cases.
In the third lecture we investigate one of the many current tiling research areas: spectral theory. Schectman made his Nobel-prize-winning discovery using diffraction analysis, and studying the mathematical version has been quite fruitful. Spectral theory of tiling dynamical systems is also of broad interest. We describe how these types of spectral analysis are carried out, give examples, and discuss what is known and unknown about the relationship between dynamical and diffraction analysis. Special attention is paid to the “point spectrum”, which is related to eigenfunctions and also to the bright spots that appear on diffraction images.
These lectures introduce the dynamical systems approach to tilings of Euclidean space, especially quasicrystalline tilings that have been constructed using a ‘supertile method’. Because tiling dynamics parallels one-dimensional symbolic dynamics, we discuss this case as well, highlighting the differences and similarities in the methods of study and the results that can be obtained.
In the first lecture we motivate the field with the discovery of ...

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## Post-edited  Bounded remainder sets for rotations on $p$-adic solenoids Haynes, Alan (Auteur de la Conférence) | CIRM (Editeur )

Bounded remainder sets for a dynamical system are sets for which the Birkhoff averages of return times differ from the expected values by at most a constant amount. These sets are rare and important objects which have been studied for over 100 years. In the last few years there have been a number of results which culminated in explicit constructions of bounded remainder sets for toral rotations in any dimension, of all possible allowable volumes. In this talk we are going to explain these results, and then explain how to generalize them to give explicit constructions of bounded remainder sets for rotations in $p$-adic solenoids. Our method of proof will make use of a natural dynamical encoding of patterns in non-Archimedean cut and project sets.
Bounded remainder sets for a dynamical system are sets for which the Birkhoff averages of return times differ from the expected values by at most a constant amount. These sets are rare and important objects which have been studied for over 100 years. In the last few years there have been a number of results which culminated in explicit constructions of bounded remainder sets for toral rotations in any dimension, of all possible allowable ...

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## Multi angle  Yet another characterization of the Pisot conjecture Mercat, Paul (Auteur de la Conférence) | CIRM (Editeur )

In the way of Arnoux-Ito, we give a general geometric criterion for a subshift to be measurably conjugated to a domain exchange and to a translation on a torus. For a subshift coming from an unit Pisot irreducible substitution, we will see that it becomes a simple topological criterion. More precisely, we define a topology on $\mathbb{Z}^d$ for which the subshift has pure discrete spectrum if and only if there exists a domain of the domain exchange on the discrete line that has non-empty interior. We will see how we can compute exactly such interior using regular languages. This gives a way to decide the Pisot conjecture for any example of unit Pisot irreducible substitution.
Joint work with Shigeki Akiyama.
In the way of Arnoux-Ito, we give a general geometric criterion for a subshift to be measurably conjugated to a domain exchange and to a translation on a torus. For a subshift coming from an unit Pisot irreducible substitution, we will see that it becomes a simple topological criterion. More precisely, we define a topology on $\mathbb{Z}^d$ for which the subshift has pure discrete spectrum if and only if there exists a domain of the domain ...

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## Multi angle  The undecidability of the domino problem Jeandel, Emmanuel (Auteur de la Conférence) | CIRM (Editeur )

One of the most fundamental problem in tiling theory is to decide, given a surface, a set of tiles and a tiling rule, whether there exist a way to tile the surface using the set of tiles and following the rules. As proven by Berger in the 60’s, this problem is undecidable in general.
When formulated in terms of tilings of the discrete plane by unit tiles with colored constraints, this is called the Domino Problem and was introduced by Wang in an effort to solve satisfaction problems for ??? formulas by translating the problem into a geometric problem.
In this course, we will give a brief description of the problem and to the meaning of the word “undecidable”, and then give two different proofs of the result.
One of the most fundamental problem in tiling theory is to decide, given a surface, a set of tiles and a tiling rule, whether there exist a way to tile the surface using the set of tiles and following the rules. As proven by Berger in the 60’s, this problem is undecidable in general.
When formulated in terms of tilings of the discrete plane by unit tiles with colored constraints, this is called the Domino Problem and was introduced by Wang in an ...

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## Multi angle  Lecture on Delone sets and tilings Solomyak, Boris (Auteur de la Conférence) | CIRM (Editeur )

In this lecture we focus on selected topics around the themes: Delone sets as models for quasicrystals, inflation symmetries and expansion constants, substitution Delone sets and tilings, and associated dynamical systems.

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## Multi angle  From combinatorial games to shape-symmetric morphisms Rigo, Michel (Auteur de la Conférence) | CIRM (Editeur )

The general aim of these lectures is to present some interplay between combinatorial game theory (CGT) and combinatorics on (multidimensional) words.
In the first introductory lecture, we present some basic concepts from combinatorial game theory (positions of a game, Nim-sum, Sprague-Grundy function, Wythoff’s game, ...). We also review some concepts from combinatorics on words. We thus introduce the well-known k-automatic sequences and review some of their characterizations (in terms of morphisms, finiteness of their k-kernel,...). These sequences take values in a finite set but the Sprague-Grundy function of a game, such as Nim of Wythoff, is usually unbounded. This provides a motivation to introduce k-regular sequences (in the sense of Allouche and Shallit) whose k-kernel is not finite, but finitely generated.
In the second lecture, games played on several piles of token naturally give rise to a multi-dimensional setting. Thus, we reconsider k-automatic and k-regular sequences in this extended framework. In particular, determining the structure of the bidimensional array encoding the (loosing) P-positions of the Wythoff’s game is a long-standing and challenging problem in CGT. Wythoff’s game is linked to non-standard numeration system: P-positions can be determined by writing those in the Fibonacci system. Next, we present the concept of shape-symmetric morphism: instead of iterating a morphism where images of letters are (hyper-)cubes of a fixed length k, one can generalize the procedure to images of various parallelepipedic shape. The shape-symmetry condition introduced twenty years ago by Maes permits to have well-defined fixed point.
In the last lecture, we move to generalized numeration systems: abstract numeration systems (built on a regular language) and link them to morphic (multidimensional) words. In particular, pictures obtained by shape-symmetric morphisms coincide with automatic sequences associated with an abstract numeration system. We conclude these lectures with some work in progress about games with a finite rule-set. This permits us to discuss a bit Presburger definable sets.
The general aim of these lectures is to present some interplay between combinatorial game theory (CGT) and combinatorics on (multidimensional) words.
In the first introductory lecture, we present some basic concepts from combinatorial game theory (positions of a game, Nim-sum, Sprague-Grundy function, Wythoff’s game, ...). We also review some concepts from combinatorics on words. We thus introduce the well-known k-automatic sequences and review ...

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## Multi angle  Dimension groups and recurrence for tree subshifts Berthé, Valérie (Auteur de la Conférence) | CIRM (Editeur )

Dimension groups are invariants of orbital equivalence. We show in this lecture how to compute the dimension group of tree subshifts. Tree subshifts are defined in terms of extension graphs that describe the left and right extensions of factors of their languages: the extension graphs are trees. This class of subshifts includes classical families such as Sturmian, Arnoux-Rauzy subshifts, or else, codings of interval exchanges. We rely on return word properties for tree subshifts: every finite word in the language of a tree word admits exactly d return words, where d is the cardinality of the alphabet.
This is joint work with P. Cecchi, F. Dolce, F. Durand, J. Leroy, D. Perrin, S. Petite.
Dimension groups are invariants of orbital equivalence. We show in this lecture how to compute the dimension group of tree subshifts. Tree subshifts are defined in terms of extension graphs that describe the left and right extensions of factors of their languages: the extension graphs are trees. This class of subshifts includes classical families such as Sturmian, Arnoux-Rauzy subshifts, or else, codings of interval exchanges. We rely on return ...

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## Multi angle  Algebraic sums and products of univoque bases Dajani, Karma (Auteur de la Conférence) | CIRM (Editeur )

Given $x\in(0, 1]$, let ${\mathcal U}(x)$ be the set of bases $\beta\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^{\infty}{{d_i}/{\beta^i}}$. In 2014, Lü, Tan and Wu proved that ${\mathcal U}(x)$ is a Lebesgue null set of full Hausdorff dimension. In this talk, we will show that the algebraic sum ${\mathcal U}(x)+\lambda {\mathcal U}(x)$, and the product ${\mathcal U}(x)\cdot {\mathcal U}(x)^{\lambda}$ contain an interval for all $x\in (0, 1]$ and $\lambda\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters associated with the family of symmetric binary expansions studied recently by the first speaker and C. Kalle.
This is joint work with V. Komornik, D. Kong and W. Li.

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