On the number of topological orbits of complex germs in classes (xy, xa + yb )

Aldicio José Miranda; Liane Mendes Feitosa Soares; Marcelo José Saia

doi:10.1017/S0308210516000111

On the number of topological orbits of complex germs in classes (xy, xa + yb )

Part of: Singularities Theory of singularities and catastrophe theory

Published online by Cambridge University Press: 27 October 2016

Aldicio José Miranda ,

Liane Mendes Feitosa Soares and

Marcelo José Saia

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Aldicio José Miranda: Affiliation:
Faculdade de Matemática, Universidade Federal de Uberlândia, Campus Santa Mônica – Bloco 1F – Sala 1F120, Av. João Naves de Avila 2121, 38.408-10 Uberlândia – MG, Brazil (aldicio@ufu.br)
Liane Mendes Feitosa Soares: Affiliation:
Centro de Ciências da Natureza, Universidade Federal do Piaui, Av. Universitária 661, 64049-550 Teresina – PI, Brazil (liane@ufpi.br)
Marcelo José Saia: Affiliation:
Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Av. Trabalhador Sancarlense 400, 13566-590 São Carlos – SP, Brazil (mjsaia@icmc.usp.br)

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We show that there exist an infinite number of topological orbits in classes of complex map germs from the plane to the plane that have a representative of type (xy, xa + yb ), with (a, b) ≠ = (2, 3) or (2, 5). Our key tool to prove this existence is the existence (or not) of stems in the class; these germs are not -finitely determined and allow the determination of a non-finite number of topological orbits. We also show that the class (xy, x 2 + y 5) has two topological orbits.