Dynamical Systems Around the Rauzy Gasket and Their Ergodic Properties

Abstract

At the beginning of the 80s, H. Masur and W. Veech started the study of generic properties of interval exchange transformations (IETs) proving that almost every such transformation is uniquely ergodic. About the same time, S. Novikov’s school and French mathematicians independently discovered very intriguing phenomena for classes of measured foliations on surfaces and respective IETs. For instance, minimality is exceptional in these families. A precise version of this statement is a conjecture by Novikov. The French and Russian constructions are very different ones. Nevertheless, in the most simple situation (surfaces of genus three with two singularities) it was recently observed that both foliations share the same type of properties. For instance, the space of minimal parameters is the same, called the Rauzy gasket. However, the precise connection between these two series of works was rather unclear. The aim of this paper is to prove that both theories describe, in different languages, the same objects. This text provides an explicit dictionary between both constructions.

Tiling billiards and Dynnikov's helicoid

“…Many other definitions have been given throughout the last fourty years, related to the circle percolation (Meester-Nowicki [21]), dynamics of Arnoux-Rauzy family of interval exchange transformations (Arnoux-Rauzy [2]), systems of isometries (Dynnikov-Skripchenko [14]), Novikov's sections of some polyhedral object (Dynnikov-DeLeo [9]), and, as shown here, in relation to the dynamics of nonlinearly escaping trajectories of tiling billiards (Davis et al [5], Hubert and ourselves [18,24]). In the last years the understanding emmerged that all these different interpretations of the Rauzy gasket are equivalent, as is implied in this work and shown in the works [13] and [18].…”

A Rauzy fractal unbounded in all directions of the plane

“…This theorem completes the works of Arnoux and Starosta, who conjectured it in 2013, to prove that the Arnoux-Rauzy continued fraction algorithm detects all kind of rational dependencies [3]. Note that it has been recently proved by Dynnikov, Hubert and Skripchenko using quadratic forms [6].…”

“…Ce résultat, conjecturé par Arnoux et Starosta en 2013 [3], a été démontré très récemment par des moyens plus sophistiqués par Dynnikov, Hubert et Skripchenko [6].…”

Renormalization in one-dimensional dynamics

“…Using Veech's suspension construction (also known as the zippered rectangle model), one can define a surface of genus 3 such that the Arnoux-Rauzy IETs appear as the first return map on a transversal to the directional flow on this surface. A detailed study of the connection between chaotic regimes in the Novikov problem and Arnoux-Rauzy IETs can be found in [27].…”