Related papers: Non-Autonomous Basins of Attraction With 4-Dimensi…

We give an interesting example of a map in $\mathbb{C}^2$ that is tangent to the identity, but that does not have a domain of attraction along any of its characteristic direction. This map has three characteristic directions, two of which…

We study the homeomorphic extension of biholomorphisms between convex domains in $\mathbb C^d$ without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between…

In many applications one is interested in finding the stability regions (basins of attraction) of some stationary states (attractors). In this paper we show that one cannot compute, in general, the basins of attraction of even very regular…

Given an n-dimensional C^r-diffeomorphism g, its renormalized iteration is an iteration of g, restricted to a certain n-dimensional ball and taken in some C^r-coordinates in which the ball acquires radius 1. We show that for any r >/- 1 the…

We announce the discovery of a diffeomorphism of a three-dimensional manifold with boundary which has two disjoint attractors. Each attractor attracts a set of positive $3$-dimensional Lebesgue measure whose points of Lebesgue density are…

We numerically explore the Newton-Raphson basins of convergence, related to the libration points (which act as attractors), in the planar circular restricted five-body problem (CR5BP). The evolution of the position and the linear stability…

A basic problem in complex dynamics is to understand orbits of holomorphic maps. One problem is to understand the collection of points $S$ in an attracting basin whose forward orbits land exactly on the attracting fixed point. In the paper…

In this paper we study the dynamics of Halley's and Traub's root-finding algorithms applied to a symmetric family of polynomials of degree $d+1\geq 3$. We discuss the (un)boundedness and simple connectivity of the immediate basins of…

We investigate boundedness of hyperbolic components in the moduli space of Newton maps. For quartic maps, (i) we prove hyperbolic components possessing two distinct attracting cycles each of period at least two are bounded, and (ii) we…

We study the automorphisms group action on a bounded domain in $\CC^n$ having a boundary point that is exponentially flat. Such a domain typically has a compact automorphism group. Our results enable us to generate many new examples.

We study the possible dimensions that the groups of holomorphic automorphisms of hyperbolic Reinhardt domains can have. We are particularly interested in the problem of characterizing Reinhardt domains with automorphism group of prescribed…

We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their…

In the vector-field guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a one-dimensional geometric desired path. The existence of singular points where the…

We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in $\mathbb{C}^n, \ n > 1$. Specifically: we prove that any proper holomorphic self-map of a certain type of balanced, finite-type domain in…

Let $f\colon \mathbb{C} \to \mathbb{C}$ be a transcendental entire map from the Eremenko-Lyubich class $\mathcal{B}$, and let $\zeta$ be an attracting periodic point of period $p$. We prove that the boundaries of components of the…

In this paper, we study the attracting basins of the origin in C^(k+1) for the polynomial lifts of Lattes examples. We show that their boundaries are obtained as quotient of a spherical hypersurface and we explicit the singularities that…

We show that special perturbations of a particular holomorphic map on $\mathbf{P}^k$ give us examples of maps that possess chaotic nonalgebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we…

We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new…

We study when a map between two subsets of a Boolean domain W can be extended to an automorphism of W. Under many hypotheses, if the underlying Boolean algebra is complete or if the sets are finite or Boolean domains, the necessary and…

It is shown that in dimension at least three a local diffeomorphism of Euclidean n-space into itself is injective provided that the pull-back of every plane is a Riemannian submanifold which is conformal to a plane. Using a similar…