Arithmetic and geometric aspects of the critical orbits in two-symbol circle maps

Wednesday, 26 April 2023

Abstract

A classical paradigm model of one-dimensional dynamics is rigid rotation on a circle by a fixed angle.  If the circle is partitioned into two segments, each labelled by a different symbol,  any orbit on the circle generates a binary symbol sequence. The complexity of this sequence relates to the irrationality properties of the rotation angle.  A critical curve is the locus of parameters (rotation angle, partition parameter) for which the boundaries of the partition that define the symbolic dynamics are connected via a prescribed number of rotations and symbolic itinerary -- the rotation angle in this case is a quasi-Sturmian irrational.  

We study the arithmetical and geometrical properties of these curves in parameter space via Farey sequences and continued fraction expansions and continuant polynomials. As well, we describe some of the historical motivation for our interest from the dynamics of a family of piecewise-linear planar maps, which are linear on each of the right and left half-planes, and were studied extensively by Lagarias and Rains.

This is joint work with Franco Vivaldi (London) and Asaki Saito (Hakodate)