Related papers: On the failure of linearization for germs of $C^1$…

We describe the equivalence classes of germs of generic $2$-parameter families of complex vector fields $\dot z = \omega_\epsilon(z)$ on $\mathbb{C}$ unfolding a singular parabolic point of multiplicity $k+1$: $\omega_0= z^{k+1}…

We study conjugacy classes of germs of non-flat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to…

We describe the equivalence classes of germs of generic 1-parameter families of complex vector fields z dot = omega_epsilon(z) on C unfolding a singular point of multiplicity k+1: omega_0 = z^{k+1} + o(z^{k+1}). The equivalence is under…

The formal class of a germ of diffeomorphism $\phi$ is embeddable in a flow if $\phi$ is formally conjugated to the exponential of a germ of vector field. We prove that there are complex analytic unipotent germs of diffeomorphisms at…

We give a classification of superattracting germs in dimension one over a complete normed algebraically closed field of positive characteristic up to conjugacy. In particular we show that formal and analytic classifications coincide for…

Let $\mathscr{X}^r(M)$ be the set of $C^r$ vector fields on a boundaryless compact Riemannian manifold $M$. Given a vector field $X_0\in\mathscr{X}^r(M)$ and a compact invariant set $\Gamma$ of $X_0$, we consider the closed subset…

We prove that a hyperbolic Dulac germ with complex coefficients in its expansion is linearizable on a standard quadratic domain and that the linearizing coordinate is again a complex Dulac germ. The proof uses results about normal forms of…

We show that if a H\"{o}lder continuous linear cocycle over a hyperbolic system is measurably conjugate to a cocycle taking values in a unipotent group, then the cocycle is H\"older continuously conjugate to a cocycle taking values in a…

In this note we announce a result for vector fields on three-dimensional manifolds: those who are singular hyperbolic or exhibit a homoclinic tangency form a dense subset of the space of $C^1$-vector fields. This answers a conjecture by…

We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field $b\colon (0,T) \times \mathbb R^d \to \mathbb R^d$, $T>0$. This class of vector fields arises in the context of hyperbolic…

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,…

Germs of tubular neighborhood embeddings for submanifolds N of manifolds M are in one-one correspondence with germs of Euler-like vector fields near N. In many contexts, this reduces the proof of `normal forms results' for geometric…

It is well-known that the flows generated by two smooth vector fields commute, if the Lie bracket of these vector fields vanishes. This assertion is known to extend to Lipschitz continuous vector fields, up to interpreting the vanishing of…

In this paper we establish a criterion for the triviality of the $C^1$-centralizer for vector fields and flows. In particular we deduce the triviality of the centralizer at homoclinic classes of $C^r$ vector fields ($r\ge 1$). Furthermore,…

In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold $M^d$ has a `small' centralizer. In the $C^1$ case, we give two criteria, one of which is $C^1$-generic, which…

We give a proof of an extension of the Hartman-Grobman theorem to nonhyperbolic but asymptotically stable equilibria of vector fields. Moreover, the linearizing topological conjugacy is (i) defined on the entire basin of attraction if the…

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…

We construct an example of a bounded degree, nonamenable, unimodular random rooted graph with $p_c=p_u$ for Bernoulli bond percolation, as well as an example of a bounded degree, unimodular random rooted graph with $p_c<1$ but with an…

We study Siegel's center problem on the linearization of germs of diffeomorphisms in one variable. In addition of the classical problems of formal and analytic linearization, we give sufficient conditions for the linearization to belong to…

We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\{(x,y,z)\in\C^3 |…