Related papers: Simultaneous linearization of holomorphic germs in…

Let $f_1, ..., f_h$ be $h\ge 2$ germs of biholomorphisms of $\C^n$ fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if $f_1, ..., f_h$ commute and their linear…

Let $f$ be a germ of holomorphic diffeomorphism of $\C^n$ fixing the origin $O$, with $df_O$ diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of $df_O$ and some restrictions on the resonances, $f$ is…

Let f_1,...,f_N be commuting germs of holomorphic diffeomorphisms in C fixing the origin with irrational rationally independent rotation numbers alpha_1,...,alpha_N. We adapt Yoccoz' renormalization of germs to this setting to show that a…

We study the linearization problem of germs of holomorphic diffeomorphisms with resonant linear part. The formal linearization requires in general an infinite number of algebraic relations to be satisfied by the coefficients of the power…

We prove the holomorphic linearizability of germs of biholomorphisms of (C n , 0), fixing the origin, point at which the linear part has nontrivial Jordan blocks under the following assumptions : We first assume the eigenvalues are of…

We describe a canonical procedure for associating to any (germ of) holomorphic self-map f of C^n fixing the origin such that df_O is invertible and non-diagonalizable an n-dimensional complex manifold M, a holomorphic map p from M to C^n, a…

We prove that the linearization of a germ of holomorphic map of the type $F_\lambda(z)=\lambda(z+O(z^2))$ has a $ C^1$--holomorphic dependence on the multiplier $\lambda$. $C^1$--holomorphic functions are $ C^1$--Whitney smooth functions,…

Suppose that $M$ is a closed isotropic Riemannian manifold and that $R_1,...,R_m$ generate the isometry group of $M$. Let $f_1,...,f_m$ be smooth perturbations of these isometries. We show that the $f_i$ are simultaneously conjugate to…

The goal of this paper is to study the dynamics of holomorphic diffeomorphisms in C^n such that the resonances among the first 1<= r<= n eigenvalues of the differential are generated over N by a finite number of Q-linearly independent…

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in C^n whose differentials have one-dimensional family of resonances in the first m eigenvalues, m <= n (but more resonances are allowed…

The classification, by topological conjugacy, of invertible holomorphic germs $f:(\mathbb{C}^n,0)\to (\mathbb{C}^n,0)$, with $\lambda_1,...,\lambda_n$ eigenvalues of $df_0$, and $|\lambda_i|\neq 1$ for $i=2,...,n$ while $\lambda_1$ is a…

Following Favre, we define a holomorphic germ f:(C^d,0) -> (C^d,0) to be rigid if the union of the critical set of all iterates has simple normal crossing singularities. We give a partial classification of contracting rigid germs in…

Let $A$ be a diagonal linear operator on $\C^n$, with all eigenvalues satisfying $0<|\alpha_i|<1$, and $M = (\C^n\backslash 0)/<A>$ the corresponding Hopf manifold. We show that any stable holomorphic bundle on $M$ can be lifted to a…

By applying holomorphic motions, we prove that a parabolic germ is quasiconformally rigid, that is, any two topologically conjugate parabolic germs are quasiconformally conjugate and the conjugacy can be chosen to be more and more near…

Suppose that $M$ is a connected orientable $n$-dimensional manifold and $m>2n$. If $H^i(M,\R)=0$ for $i>0$, it is proved that for each $m$ there is a monomorphism $H^m(W_n,\on{O}(n))\to H^m_{\on{cont}}(\on{Diff}M,\R)$. If $M$ is closed and…

In this paper we study the dynamics of germs of quasi-parabolic one-resonant biholomorphisms of $\C^{n+1}$ fixing the origin, namely, those germs whose differential at the origin has one eigenvalue 1 and the others having a one dimensional…

In this paper we study the existence of basins of attraction for germs of 2-resonant biholomorphisms of $\C^n$ fixing a point, that is germs such that the eigenvalues of the differential at the fixed point have a 2 dimensional family of…

Local analytic germs can be simultaneously deformed equivariantly for the flow if there is one holomorphic solution whose degree is high compared to local discrepancy.

Let $M$ be a $n$-dimensional complex manifold and $f,g:M\to M$ two distinct holomorphic self-maps. Suppose that $f$ and $g$ coincide on a globally irreducible compact hypersurface $S\subset M$. We show that if one of the two maps is a local…

We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point.We also give condition that ensure that such a group…