Backward Rauzy-Veech algorithm and horizontal saddle connections

TL;DR

This paper investigates how the backward Rauzy-Veech algorithm acts on translation surfaces, establishing a link between the absence of horizontal saddle connections and orbit completeness, as well as connecting saddle connections to flow minimality.

Contribution

It proves that the backward Rauzy-Veech orbit is infinite if and only if the surface lacks horizontal saddle connections, and relates saddle connection appearance to flow minimality.

Findings

Orbit is infinite iff no horizontal saddle connections exist.

Appearance of all saddle connections implies horizontal flow minimality.

Backward Rauzy-Veech algorithm characterizes surface properties.

Abstract

The goal of this note is to study the action of the backward Rauzy-Veech algorithm on the translation surfaces with horizontal saddle connections. In particular, we prove that the orbit of a translation surface via the aforementioned algorithm is $\infty$-complete if and only if the surface does not posses horizontal saddle connections. Moreover, we show that appearance of all saddle connections as sides of the polygonal representations of the translations surface along the backward Rauzy-Veech induction orbit is equivalent to the minimality of the horizontal translation flow.