For an entire function $f$ let $N\left(z\right)=z-f\left(z\right)/f^{\prime }\left(z\right)$ be the Newton function associated to $f$. Each zero $\xi$ of $f$ is an attractive fixed point of $N$ and is contained in an invariant component of the Fatou set of the meromorphic function $N$ in which the iterates of $N$ converge to $\xi$. If $f$ has an asymptotic representation $f\left(z\right)\sim exp(-z^n),n\in \mathbb{N}$, in a sector $|argz|\<\epsilon$, then there exists an invariant component of the Fatou set where the iterates of $N$ tend to infinity. Such a component is called an invariant Baker domain.A question in the opposite direction...

We study two complex invariant manifolds associated with the parabolic fixed point of the area-preserving Hénon map. A single formal power series corresponds to both of them. The Borel transform of the formal series defines an analytic germ. We explore the Riemann surface and singularities of its analytic continuation. In particular we give a complete description of the “first” singularity and prove that a constant, which describes the splitting of the invariant manifolds, does not vanish. An interpretation...

Let Q be the unit square in the plane and h: Q → h(Q) a quasiconformal map. When h is conformal off a certain self-similar set, the modulus of h(Q) is bounded independent of h. We apply this observation to give explicit estimates for the variation of multipliers of repelling fixed points under a "spinning" quasiconformal deformation of a particular cubic polynomial.

We prove real bounds for interval maps with one reflecting critical point.

Laminations are classic sets of disjoint and non-self-crossing curves on surfaces. Lamination languages are languages of two-way infinite words which code laminations by using associated labeled embedded graphs, and which are subshifts. Here, we characterize the possible exact affine factor complexities of these languages through bouquets of circles, i.e. graphs made of one vertex, as representative coding graphs. We also show how to build families of laminations together with corresponding lamination...

We construct branched coverings such as matings and captures to describe the dynamics of every critically finite cubic Newton map. This gives a combinatorial model of the set of cubic Newton maps as the gluing of a subset of cubic polynomials with a part of the filled Julia set of a specific polynomial (Figure 1).

Given a holomorphic mapping $f:{\mathbb{P}}^2\to {\mathbb{P}}^2$ of degree $d\ge 2$ we give sufficient conditions on a positive closed (1,1) current of $S$ of unit mass under which $d^{-n}{f}^{n*}S$ converges to the Green current as $n\to \infty$. We also conjecture necessary condition for the same convergence.