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Divergent diagrams of folds associated with reflections

Patrícia Hernandes Baptistelli, Maria Elenice Rodrigues Hernandes, Miriam Manoel

Abstract

We analyse divergent diagrams of \(k\)-fold map-germs on \((\mathbb{C}^n,0)\), for $k, n \geq 2$, associated with reflections, adapting to the complex setting the theory of folds associated with involutions on \((\mathbb{R}^n,0)\). In the complex case, a \(k\)-fold is naturally related to a cyclic group generated by a reflection, which guides the analytic classification of singularities. Under the conditions of transversality and linearity of the associated reflections, certain conditions related to the nontrivial eigenvalues appear as invariants by simultaneous conjugacy. We also provide a complete classification of pairs of transversal linear reflections and the corresponding divergent diagrams.

Divergent diagrams of folds associated with reflections

Abstract

We analyse divergent diagrams of -fold map-germs on \((\mathbb{C}^n,0)\), for , associated with reflections, adapting to the complex setting the theory of folds associated with involutions on \((\mathbb{R}^n,0)\). In the complex case, a -fold is naturally related to a cyclic group generated by a reflection, which guides the analytic classification of singularities. Under the conditions of transversality and linearity of the associated reflections, certain conditions related to the nontrivial eigenvalues appear as invariants by simultaneous conjugacy. We also provide a complete classification of pairs of transversal linear reflections and the corresponding divergent diagrams.
Paper Structure (6 sections, 20 theorems, 53 equations)

This paper contains 6 sections, 20 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. $k$-folds associated with reflections
  3. $k$-folds associated with reflections of order $k$
  4. Divergent diagrams of $\underline k$-folds
  5. Normal forms under transversality
  6. Normal forms of divergent diagrams associated with pairs of reflections

Key Result

Lemma 2.3

For $f_1$ given in (eq:f1), any reflection associated with $f_1$ belongs to the group $\langle \varphi_1\rangle$, with $\varphi_1$ given in (eq:varphi1).

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Example 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 33 more