2024/11/27 by Enhui Shi, Hui Xu, Shi, Enhui +3
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2411.18360
Let \mathbbA be an annulus in the plane \mathbb R2 and g:\mathbbA→ \mathbbA be a boundary components preserving homeomorphism which is distal and has no periodic points. In \citeSXY, the authors show that there is a continuous decomposition \mathcal P of \mathbbA into g-invariant circles such that all the restrictions of g on them share a common irrational rotation number (also called the rotation number of g) and all these circles are linearly ordered by the inclusion relation on the sets of bounded components of their complements in \mathbb R2. In this note, we show that if the decomposition \mathcal P above has a continuous section, then g can be linearized, that is it is topologically conjugate to a rigid rotation on \mathbbA. For every irrational number α∈ (0, 1), we show the existence of such a distal homeomorphism g on \mathbbA that it cannot be linearized and its rotation number is α.