Topological and bi-Lipschitz contact equivalences of germs of differentiable maps.

Author(s)

João Carlos Ferreira Costa

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2005-12-07

Examining board members

Maria Aparecida Soares Ruas; Lev Birbrair; Daniel Cantergiani Panazzolo; Angela Maria Sitta; João Nivaldo Tomazella

• Advisor: Maria Aparecida Soares Ruas
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Location: Universidade de São Paulo. Instituto de Ciências Matemáticas e de Computação. Biblioteca Prof. Achille Bassi; ICMSC/T; C837ec

Abstract

In this work we study the contact equivalence from the topological and bi-Lipschitz point of view. We characterize completely the real function-germs with respect to C0-equivalence, defining an invariant called tent function. Furthermore, we present a normal frorn for C0-finitely determined real analytic function-germs when the source dimension is n = 2. For map-germs ((Rn, 0) &rarr; (R>sup>p, 0), if n < p, we prove that all C0-finite germs are C0-equivalent. If n &ge; p, our main results are related to families of germs. Based upon regularity conditions on the families of zero-sets, we give sufficient conditions for the C0-triviality of families of C0-finite germs. In the special case of curves (p = n-1), we prove in some cases that the nurnber of half-branches of the curve is a complete invariant for the C0-equivalence. We introduce the definition of K-bi-Lipschitz equivalence and we study this equivalence relation for functions. Our main result shows that the nurnber of K-bi-Lipschitz equivalence classes of polynomial function-germs is finite. (AU)