Curlicues generated by circle homeomorphisms

TL;DR

This paper explores the geometric curves generated by circle homeomorphisms and links their properties to the underlying dynamical features like rotation number and cohomological equations.

Contribution

It establishes connections between geometric properties of generated curves and the dynamical characteristics of circle homeomorphisms, providing new insights into their interplay.

Findings

Boundedness of curves linked to rotation number

Superficiality related to displacement sequences

Local curvature properties connected to cohomological solutions

Abstract

We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as boundedness, superficiality, local discrete radius of curvature are linked with dynamical properties of the circle homeomorphism which generates them: rotation number and its continued fraction expansion, existence of a continuous solution of the corresponding cohomological equation and displacement sequence along the orbit.