The Wild, Elusive Singularities of the T-fractal Surface

TL;DR

This paper rigorously defines the T-fractal translation surface, explores its geometric and dynamical properties, and introduces the concept of elusive singularities, which generalize wild singularities with complex rotational structures.

Contribution

It provides a formal definition of the T-fractal surface and characterizes its elusive singularities, extending the understanding of wild singularities in fractal translation surfaces.

Findings

Existence of a Cantor set of elusive singularities.

Elusive singularities have infinite discrete sets of rotational components.

Rotational components with irrational addresses have zero length.

Abstract

We give a rigorous definition of the T-fractal translation surface, and describe some its basic geometric and dynamical properties. In particular, we study the singularities attached to the surface by its metric completion and show there exists a Cantor set of "elusive singularities." We show these elusive singularities can be thought of as a generalization of the wild singularities introduced by Bowman and Valdez. In particular, we show that every elusive singularities has an infinite discrete set of rotational components. We also show that each rotational component of an elusive singularity with an irrational address has zero length (angle).