Multidimensional Continued Fractions and Euclidean Dynamics, 23-27 March 2020

This workshop was initially supposed to be held at Lorentz Center Workshop site

This workshop is devoted to ergodic properties of Markovian multidimensional continued fraction algorithms by bringing together leading specialists working in various related areas of dynamical systems such as Teichmüller dynamics, hyperbolic dynamics, homogeneous dynamics, lattice reduction algorithms, symbolic dynamics, Diophantine approximation, etc. The main targeted questions concerning continued fractions are invariant measures, convergence properties and Lyapunov exponents, and impact of the dimension.

Scientific organizers: Valérie Berthé, Konstantin Khanin, Alexandra Skripchenko, Evgeny Verbitskiy.

Monday March 23

Title: Homogeneous Methods in Number Theory

Abstract: We give a brief example of how to use ergodic theory in Diophantine approximation, using only the Birkhoff ergodic theorem. This is a part of joint work with Parrish and Tseng.

Title: Simplicity of Lyapunov exponents for integral cocycles over countable shifts

Abstract: The study of certain features of many multidimensional continued fraction algorithms (including Rauzy induction for interval exchange maps, Rauzy algorithm for special systems of isometries, fully subtractive algorithms, ...) are often related to the Lyapunov exponents of integral cocycles over countable shifts. In particular, it is sometimes important to determine the positivity, negativity and/or simplicity of such Lyapunov exponents.

In this talk, we discuss a Galois-theoretical criterion (from a joint work with Moeller and Yoccoz) of simplicity of Lyapunov exponents of integral cocycles over countable shifts. A curious feature of this simplicity criterion is that it is particularly well-suited for concrete calculations and/or numerical experiments because it is based on a finite computation with the Galois groups and splitting fields of characteristic polynomials of integral matrices. In particular, this criterion was recently successfully employed by Avila-Hubert-Skripchenko in relation to the Rauzy gasket (and Novikov's problem) and Fougeron-Skripchenko in relation to the triangle sequence algorithm and the Cassaigne algorithm, and we hope that this criterion might be helpful in several other contexts."

  • 4pm Discussion time with Carlos Matheus and Jayadev Athreya

Tuesday March 24

Title: Geometry and dynamics of best approximation

Abstract: In this talk I will explain several ways the theory of continued fractions can be generalized to higher dimensions based on the concept of best approximation, their connections to other more classical formulations, such as the Minkowski continued fraction algorithm, and reformulations of the Littlewood Conjecture and the Lonely Runner Conjecture in this framework.

Title: Minimal vectors in lattices over Gauss integers in C^2

Abstract: The minimal vector sequences in lattices over Gauss integers in C ^ 2 can be seen as a complex continued fraction expansion. This sequence is related to the first return map in a manifold T transverse to the diagonal flow in the space of unimodular Gauss lattices in C^2. The transversal T consists of lattices of which the two complex minima are equal. A complete description of T is possible and the first return can be computed by an algorithm of bounded complexity.

  • 10am Meeting time with Yitwah Cheung and Nicolas Chevallier
  • Recording

Wednesday March 25

Title: On Cross Sections to the Horocycle and Geodesic Flows on Quotients by Hecke Triangle Groups $G_q$

Abstract: The Hecke triangle groups $G_q$ are well known analogues of the modular group $SL(2, Z)$. In this context, the $G_q$-analogues of the set of visible integer points $Z_\text{prim}^2$ are the discrete linear orbits $G_q (1, 0)^T$. Euclidean algorithms for $G_q$ can be interpreted as algorithms for generating the elements of $G_q (1, 0)^T$ in planar sectors, and are coincidentally related to several cross sections to the geodesic flow on $SL(2, R) / G_q$.

In this presentation, we show how one family of $G_q$-Euclidean algorithms can be truncated to give rise to novel \emph{next-vector} algorithms for generating the elements of $G_q (1, 0)^T$ in planar strips. We discuss how those next-vector algorithms are related to particular cross sections to the horocycle flow on $SL(2, R) / G_q$, with specific maps (the \emph{BCZ maps}) as first return maps. We also discuss how several $G_q$-Farey statistics can be derived using the $G_q$-BCZ maps. Finally, we present some open questions and future directions of research.

Title: Some dynamical systems associated with continued fractions and S-adic systems

Abstract: In this lecture, we show some curious dynamical systems related to continued fractions, and to some classical constructions; we end by commenting some videos showing 3 different views of the geodesic flow on the modular surface; these videos, due to Edmund Harriss, can be found at the following addresses: Video in english Video in french show some periodic orbits of the geodesic flow. The following videos Video1 Video2 Video3 show non-periodic orbits, and can be customized to show different starting points, and also the horocycle flow.

  • 2pm Meeting time with Pierre Arnoux and Dia Taha
  • Recording

Thursday March 26

Title: On the second Lyapunov exponent of some MCF algorithms

Abstract: We study the strong convergence of certain multidimensional continued fraction algorithms. In particular, in the two-dimensional case, we prove that the second Lyapunov exponent of Selmer's algorithm is negative and bound it away from zero. Moreover, we give heuristic results on several other continued fraction algorithms. Our results indicate that all classical multidimensional continued fraction algorithms cease to be strongly convergent for high dimensions. The only exception seems to be the Arnoux-Rauzy algorithm which, however, is defined only on a set of measure zero.

This is joint work with Valérie Berthé and Jörg Thuswaldner.

  • 4pm Meeting time with Wolfgang Steiner and discussion on numerical experiments
  • Recording
  • Computer experiments by S. Labbé Post

Friday March 27

Title: Convergence problem for MCF: historical review and open questions

Title: Rauzy induction and Multidimensional continued fractions

Abstract: I will give a short introduction to Rauzy induction and its applications to Teichmüller dynamics. In a second half of the lecture I will explain how these methods adapt to the case of multidimensional continued fractions algorithms.

  • 2pm Meeting time with Sacha Skripchenko and Charles Fougeron; discussion on the convergence of MCF algorithms (different approaches and partial results)
  • Recording