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July  2005, 12(4): 657-674. doi: 10.3934/dcds.2005.12.657

Divergent diagrams of folds and simultaneous conjugacy of involutions

Solange Mancini 1, , Miriam Manoel 2, and  Marco Antonio Teixeira 3,

1. 

Departamento de Matemática, IGCE, Universidade Estadual Paulista, 13500-230 Caixa Postal 178, Rio Claro, SP, Brazil
2. 

Departamento de Matemática, ICMC, Universidade de São Paulo, 13560-970 Caixa Postal 668, São Carlos, SP, Brazil
3. 

Departamento de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, S.P., Brazil

Received  November 2003 Revised  September 2004 Published  January 2005

In this work we show that the smooth classification of divergent diagrams of folds $(f_1, \ldots, f_s) : (\mathbb R^n,0) \to (\mathbb R^n \times \cdots \times \mathbb R^n,0)$ can be reduced to the classification of the $s$-tuples $(\varphi_1, \ldots, \varphi_s)$ of associated involutions. We apply the result to obtain normal forms when $s \leq n$ and $\{\varphi_1, \ldots, \varphi_s\}$ is a transversal set of linear involutions. A complete description is given when $s=2$ and $n\geq 2$. We also present a brief discussion on applications of our results to the study of discontinuous vector fields and discrete reversible dynamical systems.
Keywords: singularities, involution, discontinuous vector fields, reversible diffeomorphisms., Divergent diagram of folds, normal form.
Mathematics Subject Classification: 34G10, 47D06, 58F11, 92D2.
Citation: Solange Mancini, Miriam Manoel, Marco Antonio Teixeira. Divergent diagrams of folds and simultaneous conjugacy of involutions. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 657-674. doi: 10.3934/dcds.2005.12.657
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