Teichm\"uller dynamics, dilation tori and piecewise affine circle homeomorphisms

TL;DR

This paper explores the geometry and dynamics of dilation tori with two singularities, showing that their vertical foliations are almost always Morse-Smale, and applies this to generic piecewise affine circle homeomorphisms.

Contribution

It establishes the Morse-Smale property for the vertical foliation of dilation tori and generic piecewise affine circle homeomorphisms, linking Teichmuller dynamics to these properties.

Findings

Vertical foliation of dilation tori is almost always Morse-Smale.

Generic piecewise affine circle homeomorphisms are Morse-Smale.

Provides a geometric framework connecting Teichmuller flow and circle homeomorphisms.

Abstract

We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic piecewise affine circle homeomorphism with two break points -with respect to the Lebesgue measure- is Morse-Smale.