arXiv subjects

Emanoel Souza

Publications and source records attributed to Emanoel Souza.

5 recordsLinked to original sources

arXiv2025

On the minimality of pancake decomposition of surface germs

The abnormal surfaces called snakes and circular snakes, defined in \cite{GabrielovSouza}, are special types of surface germs capturing the outer Lipschitz phenomena relevant to the outer classification problem. We provide algorithms to obtain a minimal pancake decomposition, i.e., where the number of pancakes is minimal, for snakes and circular snakes. We call a pancake decomposition obtained from our algorithm a greedy pancake decomposition. We also prove that greedy pancake decompositions of weakly outer Lipschitz equivalent snakes (or circular snakes) are weakly equivalent, in the sense that there is a weakly outer bi-Lipschitz homeomorphism between the surfaces mapping each greedy pancake to a greedy pancake. This implies that such minimal decompositions are also canonical up to weakly outer bi-Lipschitz equivalence.

math.MG↗

arXiv2025

On the Invariance of the Real Milnor Number under Asymptotically Lipschitz Equivalence

We investigate sufficient conditions for the invariance of the real Milnor number under $\mathcal{R}$-bi-Lipschitz equivalence for function-germs $ f, g \colon (\mathbb{R}^n, 0) \to (\mathbb{R}, 0) $. More generally, we explore its invariance within the extended framework of $\mathcal{R}$-asymptotically Lipschitz equivalence. To this end, we introduce the $α$-derivative, which provides a natural setting for studying asymptotic growth. Additionally, we discuss the implications of our results in the context of $C^k$ and $C^{\infty}$ equivalences, establishing sufficient conditions for the real Milnor number to remain invariant.

math.AG↗

arXiv2025

Lipschitz geometry and combinatorics of circular snakes

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 Hölder 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.

math.MG↗

arXiv2024

Moderately discontinuous homology of real surfaces

The Moderately Discontinuous Homology (MD-Homology, for short) was created recently in 2022 by Fernández de Bobadilla at al. and it captures deep Lipschitz phenomena. However, to become a definitive powerful tool, it must be widely comprehended. In this paper, we investigate the MD-Homology of definable surface germs for the inner and outer metrics. We completely determine the MD-Homology of surfaces for the inner metric and we present a great variety of interesting MD-Homology of surfaces for the outer metric, for instance, we determine the MD-Homology of some bubbles, snake surfaces, and horns. Furthermore, we explicit the diversity of MD-Homology of surfaces for the outer metric in general, showing how hard it is to completely solve the outer classification problem. On the other hand, we show that, under specific conditions, the weakly outer Lipschitz equivalence determines completely the MD-Homology of surfaces for the outer metric, showing that these two subjects are quite related.

math.MG↗

arXiv2021

Lipschitz geometry and combinatorics of abnormal surface germs

We study outer Lipschitz geometry of real semialgebraic or, more general, definable in a polynomially bounded o-minimal structure over the reals, surface germs. In particular, any definable Hölder triangle is either Lipschitz normally embedded or contains some "abnormal" arcs. We show that abnormal arcs constitute finitely many "abnormal zones" in the space of all arcs, and investigate geometric and combinatorial properties of abnormal surface germs. We establish a strong relation between geometry and combinatorics of abnormal Hölder triangles.

math.MG↗