2017/02/20 by Ferenczi, Sébastien, Pascal Hubert, Hubert, Pascal
Computer Science · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems
paper · doi:10.48550/arxiv.1702.05989
openalex publication_date 2017/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction θ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if tan θ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and tan θ has bounded partial quotients, the square-tiled interval exchange transformation T is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations.