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Topological classification of families of diffeomorphisms without small divisors

About this Title

Javier Ribón, UFF, Instituto de Matemática, Valonguinho, Rua Mário Santos Braga, s/n, Centro, Niterói, RJ - Brasil 24020-140

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 207, Number 975
ISBNs: 978-0-8218-4748-0 (print); 978-1-4704-0589-2 (online)
Published electronically: April 30, 2010
MathSciNet review: 2676138
Keywords:Diffeomorphisms, topological classification, bifurcation theory, normal form, structural stability, tangent to the identity germs of diffeomorphism.
MSC: Primary 37C15; Secondary 37F45, 37F75, 37G05, 37G10

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Table of Contents


  • Preface
  • Chapter 1. Outline of the monograph
  • Chapter 2. Flower type vector fields
  • Chapter 3. A clockwork orange
  • Chapter 4. The $T$-sets
  • Chapter 5. The long limits
  • Chapter 6. Topological conjugation of (NSD) vector fields
  • Chapter 7. Families of diffeomorphisms without small divisors
  • Chapter 8. Topological invariants of (NSD) diffeomorphisms
  • Chapter 9. Tangential conjugations


We give a complete topological classification for germs of one-parameter families of one-dimensional complex analytic diffeomorphisms without small divisors. In the non-trivial cases the topological invariants are given by some functions attached to the fixed points set plus the analytic class of the element of the family corresponding to the special parameter. The proof is based on the structure of the limits of orbits when we approach the special parameter.

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