STABLE MANIFOLDS OF TWO-DIMENSIONAL BIHOLOMORPHISMS ASYMPTOTIC TO FORMAL CURVES

Lorena López-Hernanz (1) , Jasmin Raissy (2) , Javier Ribón (3) , Fernando Sanz-Sánchez (4)

1 Departamento de Fisica y Matematicas Universidad de Alcalà
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 UFF - Universidade Federal Fluminense [Rio de Janeiro]
4 UVa - Universidad de Valladolid [Valladolid]

Let F ∈ Diff(C 2 , 0) be a germ of a holomorphic diffeomorphism and let Γ be an invariant formal curve of F. Assume that the restricted diffeomorphism F | Γ is either hyperbolic attracting or rationally neutral non-periodic (these are the conditions that the diffeomorphism F | Γ should satisfy, if Γ were convergent, in order to have orbits converging to the origin). Then we prove that F has finitely many stable manifolds, either open domains or parabolic curves, consisting of and containing all converging orbits asymptotic to Γ. Our results generalize to the case where Γ is a formal periodic curve of F .

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Lorena López-Hernanz, Jasmin Raissy, Javier Ribón, Fernando Sanz-Sánchez. STABLE MANIFOLDS OF TWO-DIMENSIONAL BIHOLOMORPHISMS ASYMPTOTIC TO FORMAL CURVES. 2017. ⟨hal-01614768⟩