We show how to construct 'simple' symbolic dynamical systems in terms of renormalisation schemes associated with multidimensional continued fractions. Continued fractions are used here to generate infinite words thanks to the iteration of infinite sequences of substitutions. Simple means that these symbolic systems have a linear number of factors of a given length, or that they have pure discrete spectrum, or else, that they have a low symbolic discrepancy. We also discuss the relation between these notions.[-]

Mapping classes of surfaces of finite type have been classified by Nielsen and Thurston. For surfaces of infinite type (e.g. surfaces of infinite genus), no such classification is known. I will talk about the difficulties that arise when trying to generalize the Nielsen-Thurston classification to infinite-type surfaces and present a first result in this direction, concerning maps which - loosely speaking - do not show any pseudo-Anosov behavior. Joint work with Mladen Bestvina and Jing Tao.[-]

The shear coordinate is a countable coordinate system to describe increasing self-maps of the unit circle, which is furthermore invariant under modular transformations. Characterizations of circle homeomorphism and quasisymmetric homeomorphisms were obtained by D. Šarić. We are interested in characterizing Weil-Petersson circle homeomorphisms using shears. This class of homeomorphisms arises from the Kähler geometry on the universal Teichmüller space.
For this, we introduce diamond shear which is the minimal combination of shears producing WP homeomorphisms. Diamond shears are closely related to the log-Lambda length introduced by R. Penner, which can be viewed as a renormalized length of an infinite geodesic. We obtain sharp results comparing the class of circle homeomorphisms with square summable diamond shears with the Weil-Petersson class and Hölder classes. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.
This talk is based on joint work with Dragomir Šarić and Catherine Wolfram. See https://arxiv.org/abs/2211.11497.[-]

In this talk, I will discuss results obtained with Yvan Dynnikov, Paul Mercat, Olga Paris-Romaskevich and Sasha Skripchenko. Novikov's conjecture for foliations states that the restriction of a linear foliation to a triply periodic surface is generically periodic or integrable (leaves stay at bounded distance from a line). I will explain some results on families of interval exchange transformations with flips. This approach gives a partial solution to the conjecture in a non-trivial case.[-]

The applications of renormalization ideas in Dynamical Systems became increasingly popular after 1979, and, since then, they played an important role in the study of several classes of low-dimensional systems.Very roughly speaking, the philosophy of renormalization is that, after appropriate rescalings, the long time behaviors at short scales of certain systems are dictated by other systems within a fixed class S of systems. In particular, such a renormalization procedure can iterated and, as it turns out, the phrase portraits of those systems whose successive renormalizations tend to stay in a compact portion of S can often be reasonably described (”plough in the dynamical plane to harvest in the parameter space”, A. Douady).In this minicourse, we shall illustrate these ideas by explaining the com-mon strategy of ”recurrence of renormalization to compact sets” behind two different results:
1.the solutions of Masur and Veech in 1982 to Keane's conjecture of unique ergodicity of almost all interval exchange transformations;
2. the solution of Moreira–Yoccoz in 2001 to Palis' conjecture on the prevalence of stable intersections of pairs of dynamical Cantor sets whose Hausdorff dimensions are large.[-]

In these lectures I will describe many of the experiments I have done with inner and outer billiards. These include: McBilliards: a program which investigates billiards in triangles (joint with Pat Hooper)Billiard King: a program I used to solve the Moser-Neumann problemSome variants of Billiard king, which treat other outer billiards sys-tems.Billiard King3D: A program I have been using lately to study billiards in orthoschemes (a kind of tetrahedron).A program I wrote to study symplectic billiards with Sergei Tabach-nikov . I will try to show in the talks how almost all the results I have got on this topic have come from playing with these interfaces. I am open to sharing the software and helping people set up their own programs.[-]

The course will explore several related topics in number theory with dynamical and/or geometric facets: continued fractions, Diophantine approximation, and Apollonian circle packings. We will focus on both theoretical and experimental tools, a parallel goal will be to experience the role of visualization and illustration in mathematical research. In covering background material, the approach will emphasize the visual and dynamical:
(1) Continued fractions, quadratic forms, and Diophantine approximation.
(2) Hyperbolic geometry, Minkowski space, and Kleinian groups.
With these tools at hand, we will study some areas of current research:
(1) The geometry of Diophantine approximation and continued fractions in the complex plane,including algebraic starscapes and Schmidt arrangements.
(2) Apollonian circle packings, with an emphasis on their surprising relationships to the preceding topics.[-]

The course will explore several related topics in number theory with dynamical and/or geometric facets: continued fractions, Diophantine approximation, and Apollonian circle packings. We will focus on both theoretical and experimental tools, a parallel goal will be to experience the role of visualization and illustration in mathematical research. In covering background material, the approach will emphasize the visual and dynamical:
(1) Continued fractions, quadratic forms, and Diophantine approximation.
(2) Hyperbolic geometry, Minkowski space, and Kleinian groups.
With these tools at hand, we will study some areas of current research:
(1) The geometry of Diophantine approximation and continued fractions in the complex plane,including algebraic starscapes and Schmidt arrangements.
(2) Apollonian circle packings, with an emphasis on their surprising relationships to the preceding topics.[-]