Arithmetic and Spectral Structures in Aperiodic Order

Yasushi Nagai (Thursday 9:30 - 10:15)

Title: The generalized substitutions

Abstract: Substitutions or inflation rules have been studied from various point of view. Dotera-Bekku-Ziherl defined a substitution rule which is a little different from the known examples. We consider it as an example of “generalized substitutions”, for which the substitution matrices may have fractions. We investigate the meaning of such substitution matrices and the renormalization equation for generalized substitutions, and see how the latter can be used to study diffraction spectra for self-affine tilings.

Jan Mazáč (Thursday 10:45 - 11:30 )

Title: Some interesting properties of n-point correlation functions for TM and RS substitutions (slides)

Abstract: In this talk, we recall the definition of a correlation function for a (bi)infinite word over a binary alphabet, and we provide a natural extension to an n-point correlation function. We derive some useful symmetries of these functions, and together with the renormalisation relations, we use them to describe the n-point correlation functions. We show some new results by M. Coons and M. Baake on the Thue-Morse substitution, and we show that in the case of Rudin-Shapiro substitution, the situation becomes less obvious.

Maryam Hosseini (Thursday 13:30 - 14:15)

Title: Bratteli Diagram and Topological Factoring of Cantor Minimal Systems (slides)

Abstract: In this talk I will talk about the topological and algebraic rank of Cantor minimal systems and its connection with topological factoring between two systems.

Gandhar Joshi (Thursday 14:15 - 15:00)

Title: Automatic sequences and reverse reading automatons (slides)

Abstract: (Work under the supervision of Dr. Reem Yassawi) We describe the vocabulary in use; introduce to Cobham’s little theorem and Eilenberg’s theorem which are important to our study; explain how to construct and read a reverse reading automaton; and finally the main result along with some interesting consequences and some immediate future research questions we are currently working on.

Anna Klick (Thursday 15:30 - 16:15)

Title: Arithmetic Progressions in Model and Meyer Sets (slides)

Abstract: We discuss the existence of arbitrarily long arithmetic progressions in fully Euclidean model and Meyer sets.

Petra Staynova (Friday 9:30 - 10:15)

Title: How Long can Arithmetic Progressions in Automatic Sequences be? (slides)

Abstract: It is well known that the Thue-Morse sequence cannot contain infinite arithmetic progressions of 0's or 1's. However, can it contain arbitrarily long ones? And if there is a maximum length to such progressions, what is it? We explore these questions in the setting of binary automatic sequences, and provide some exact values, as well as upper bounds. This talk is based on joint work with Ibai Aedo, Neil Manibo, Yasushi Nagai, and Uwe Grimm.

Daniel Luz (Friday 10:45 - 11:30)

Title: The spectrum of some dynamical systems of algebraic origin (slides)

Abstract: We use the representation of the dynamical system as a cut-and-project scheme to obtain their dynamical spectrum, by calculating the dual cut-and-project scheme and evaluating its lattice on the Fourier module.

Álvaro Bustos (Friday 13:30 - 14:15)

Title: B-free number-theoretical shift spaces and their symmetries (slides)

Abstract: Factorisation plays a key role in number theory, being the origin of the idea of prime numbers and motivating several constructions that generalise the rich structure of factorisation in the integers to more general contexts; it is thus not surprising that sets of numbers defined by imposing restrictions on their factorisation have deep and interesting properties. We discuss (mostly two-dimensional) shift spaces constructed from such sets, which serve as a tool to study their local structure and behaviour; examples include the d-dimensional shift of visible lattice points and the family of k-free shift spaces, the latter being deeply intertwined with ideas from algebraic number theory. These shift spaces exhibit interesting (and, from certain perspectives, unusual) properties, including the combination of high complexity (positive entropy) with symmetry rigidity (the automorphism group is "essentially trivial", containing only shift maps). Our discussion will be focused, thus, on a geometrical interpretation of the notion of "symmetry" in these systems, for which the most appropriate tool is the extended symmetry group, a mild generalisation of the automorphism group which exhibits a wide variety of interesting and non-trivial behaviours in this context, in contrast to the standard automorphism group.

Ibai Aedo (Friday 14:15 - 15:00)

Title: Arithmetic Progressions in Automatic Sequences (slides)

Abstract: In this talk we address the problem of bounding, for each positive integer d, the length A(d) of the longest monochromatic arithmetic progressions of difference d appearing in aperiodic automatic sequence. We show that, for a primitive bijective automaton, there is a sequence of differences d along which A(d) grows at least polynomially in d. This result can be strengthened if the automaton is Abelian. We also consider the Rudin-Shapiro sequence, as an example of a non-bijective automatic sequence. This work is joint research with Uwe Grimm, Neil Mañibo, Yasushi Nagai and Petra Staynova.