Évènement

Eden Delight, Pierre Arnoux, Dan Rust, Andrew Mitchell, Tony Samuel

Date(s) : 29/10/2024 - 30/10/2024   iCal
11h00 - 16h00

Workshop on Random Substitutions
Organised by :
Eden Delight P. Miro
(Ateneo de Manila University)
And
Pierre Arnoux
(Université d’Aix-Marseille)
Tuesday, October 29
11:00 – Eden Delight P. Miro: Random substitutions and
their associated dynamical systems
14:00 – Dan Rust: Periodicity and automorphism
groups for random substitutions
15:00 – Andrew Mitchell: Complexity and topological
entropy of random substitution shifts
Wednesday, October 30
10:00 – Tony Samuel: Complexity and geometry of
aperiodic systems
11:00 – Pierre Arnoux: Higher dimensional extensions of
substitutions
14:00 – Open problems, research questions : Discussion

Abstracts

Eden Delight Miro
Department of Mathematics, Ateneo de Manila University, Katipunan Avenue, 1108, Quezon City, Philippines
Title: Random substitutions and their associated dynamical systems
Abstract: One way to construct interesting symbolic dynamical systems is via substitution rules
over a finite alphabet, whereby each letter of the alphabet is assigned a predetermined word. A
natural generalization of this is to consider maps called random substitutions that assign to each
letter a finite set of possible images with a probability distribution. In this talk, I will discuss some
basic dynamical and topological properties of the subshifts arising from random substitutions.

Dan Rust
School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, UK
Title: Periodicity and automorphism groups for random substitutions
Abstract: Perhaps the most basic question for a symbolic dynamical system is whether it
includes periodic points or not. Amazingly, a full classification of random substitutions that admit
periodic points is still missing. In this talk, I’ll present some of the basic results in this direction,
including criteria for determining the existence or non-existence of periodic points and
algorithms that apply to a large family of random substitutions for finding periodic points if they
exist, and therefore allow us to enumerate them. I’ll also discuss the symmetry groups for
random substitutions, which are very different from the automorphism groups for deterministic
substitutions. For instance, the automorphism groups for most random substitution subshfits
contain every finite group as a subgroup. Nevertheless, similar tools are used in their study;
namely using the hierarchical structure of words.

Andrew Mitchell
School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, UK
Title: Complexity and topological entropy of random substitution shifts
Abstract: In contrast to their deterministic counterparts, random substitution shifts often have
positive topological entropy. In this talk, I’ll provide an overview of some recent developments in
the theory of topological entropy for random substitution shifts. These include necessary and
sufficient conditions under which a primitive random substitution shift has positive entropy and a
systematic approach to calculating topological entropy, which can be leveraged whenever the
random substitution has a well-defined inflation length. I’ll also discuss how (non-primitive)
random substitutions give rise to a broad variety of shifts with sub-exponential complexity,
including intermediate growth.

Tony Samuel
Mathematics and Statistics, University of Exeter, North Park Road, Exeter, UK, EX4 4QF
Title: Complexity and geometry of aperiodic systems: Rauzy Fractals and Rauzy measures
Abstract: Aperiodic sequences and sequence spaces form prototypical mathematical models of
quasicrystals. The most quintessential examples include subshifts of Sturmian words and
substitutions, which are ubiquitous objects in ergodic theory and aperiodic order. Two of the
most striking features these shift spaces have, are that they have zero topological entropy and
are uniquely ergodic. Random substitutions are a generalisation of deterministic substitutions,
and in stark contrast to their deterministic counterparts, subshifts of random substitutions often
have positive topological entropy and exhibit uncountably many ergodic measures.
We will begin by talking about subshifts generated by Sturmian words and ways to measure
their complexity beyond topological entropy, and show how this measure of complexity can be
used to build a classification via Jarník sets. We will then build a bridge between these subshifts
and subshifts of random substitutions. We will conclude with some recent dynamical results on
subshifts of random substitutions and ways to visualise these subshifts. Namely, we will present
a method to build a new class of Rauzy fractals and Rauzy measures.

Pierre Arnoux
Institut de Mathématiques de Marseille, Université d’Aix-Marseille, Campus de Luminy, Case 907, 13288 Marseille Cedex 9
Title: Higher dimensional extensions of substitutions and their dual maps
Abstract: Words on a final alphabet of size d can be considered as coding broken lines in a
space of dimension d, with vertices the abelianisation of all the prefixes of the word. Usual
substitutions can then be interpreted as acting on broken lines, and in the case of Pisot
substitutions, this can be seen as an example of a model set. To extend the theory beyond the
Pisot case, one would need to extend to stepped surfaces, or more generally to k-dimensional
stepped manifolds.
We will show how this can be achieved, giving a formalism for k-dimensional extensions
of substitutions (or more generally, free groups endomorphisms) and their dual maps. This
allows us to study not only Pisot substitutions, but more generally hyperbolic endomorphisms of
free groups. We will show some examples, and a number of open questions. This formalism can
be combined with the S-adic framework and the idea of random substitutions.

Emplacement
I2M Luminy - TPR2, Salle de Séminaire 304-306 (3ème étage)

Catégories

• Workshop