Lipschitz geometry and combinatorics of circular snakes

Abstract

This paper explores the Lipschitz geometric and combinatorial properties of germs of real semialgebraic surfaces (or, more generally, definable in a polynomially bounded o-minimal structure) with circular link (homeomorphic to the circle $\mathbb{S}^1$). We define and investigate the outer Lipschitz geometry of the so-called circular snakes, showing what results in the paper "Lipschitz geometry and combinatorics of abnormal surface germs" (by Andrei Gabrielov and Emanoel Souza) valid to snakes still holds for the circular case. We prove the existence of a canonical decomposition for the Valette link of a circular snake into finitely many segments and nodal zones and establish some necessary and sufficient criteria to determine when it is possible to obtain a snake from a circular snake by "removing" either one of its segments or a H\"older triangle whose Valette link is contained in one of its nodal zones. We construct a combinatorial object associated with a circular snake and prove a realization theorem for this combinatorial object. We also present a weakly outer Lipschitz classification for circular snakes. Finally, we show some results about the combinatorics of binary circular snakes, which is wildly distinct from the corresponding case shown in the work of Gabrielov and Souza.