Characteristic directions of two-dimensional biholomorphisms

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

Parabolic curves of diffeomorphisms asymptotic to formal invariant curves

We prove that ifFis a tangent to the identity diffeomorphism at0\in\mathbb{C}^{2}and Γ is a formal invariant curve ofFwhich is not contained in the set of fixed points then there exists a parabolic curve (attracting or repelling) ofFasymptotic to Γ.

On Ecalle-Hakim 's theorems in holomorphic dynamics

This chapter provides detailed proofs for the results by M. Hakim regarding the dynamics of germs of biholomorphisms tangent to the identity of order k + 1 ≥ 2 and fixing the origin. One of the main questions in the study of local discrete holomorphic dynamics, i.e., in the study of the iterates of a germ of a holomorphic map of ℂᵖ at a fixed point, which can be assumed to be the origin, is when it is possible to holomorphically conjugate it to a “simple” form, possibly its linear term. It turns out that the answer to this question strongly depends on the arithmetical properties of the eigenvalues of the linear term of the germ.

Index theorems for meromorphic self-maps of the projective space

This chapter uses techniques from the theory of local dynamics of holomorphic germs tangent to the identity to prove three index theorems for global meromorphic maps of projective space. More precisely, the chapter seeks to prove a particular index theorem: Let f : ℙⁿ ⇢ ℙⁿ be a meromorphic self-map of degree ν‎ + 1 ≥ 2 of the complex n-dimensional projective space. Let Σ‎(f) = Fix(f) ∪ I(f) be the union of the indeterminacy set I(f) of f and the fixed points set Fix(f) of f. Let Σ‎(f) = ⊔subscript Greek Small Letter AlphaΣ‎subscript Greek Small Letter Alpha be the decomposition of Σ‎ in connected components, and denote by N the tautological line bundle of ℙⁿ. After laying out the statements under this theorem, the chapter discusses the proofs.