Mario Ponce

Local dynamics for fibred holomorphic transformations

Fibred holomorphic dynamics are skew-product transformations F(θ, z) = (θ + α, fθ(z)) over an irrational rotation, such that fθ is holomorphic for every θ. In this paper we study such a dynamics in a neighbourhood of an invariant curve. We obtain some results analogous to the results in the non-fibred case. In particular, we prove a fibred version of the folklore result stating that Lyapounov stability is equivalent to linearization around a fixed point. We also obtain a fibred version of the Perez-Marco continua.

On Equidistant Sets and Generalized Conics: The Old and the New

Abstract This article is devoted to the study of classical and new results concerning equidistant sets, both from the topological and metric point of view. We include a review of the most interesting known facts about these sets in Euclidean space and we prove two new results. First, we show that equidistant sets vary continuously with their focal sets. We also prove an error estimate result about approximative versions of equidistant sets that should be of interest for computer simulations. Moreover, we offer a viewpoint in which equidistant sets can be thought of as a natural generalization for conics. Along these lines, we show that the main geometric features of classical conics can be retrieved from more general equidistant sets.

On the dynamics of nonreducible cylindrical vortices

We study the dynamics of Euclidean isometric extensions of minimal homeomorphisms of compact metric spaces. Under a general hypothesis of homogeneity for the base space, we show that these systems are never minimal, thus extending a classical result of Besicovitch concerning cylindrical cascades. Moreover, using Anosov-Katok type methods, we construct a topologically transitive isometric extension over an irrational rotation with a 2-dimensional fiber. MSC: 37B05, 37E30, 37F50,

A Livšic type theorem for germs of analytic diffeomorphisms

We deal with the problem of the validity of Livsics theorem for cocycles of diffeomorphisms satisfying the orbit periodic obstruction over an hyperbolic dynamics. We give a result in the positive direction for cocycles of germs of analytic diffeomorphisms at the origin.

On the Persistence of Invariant Curves for Fibered Holomorphic Transformations

We consider the problem of the persistence of invariant curves for analytical fibered holomorphic transformations. We define a fibered rotation number associated to an invariant curve. We show that an invariant curve with a prescribed fibered rotation number persists under small perturbations on the dynamics provided that the pair of rotation numbers verifies a Brjuno type arithmetical condition. Nevertheless, an extra complex parameter is added to the problem and the persistence becomes a one-complex codimension property.

Towards a semi-local study of parabolic invariant curves for fibered holomorphic maps

We introduce the study of the local dynamics around a parabolic indifferent invariant curve for fibred holomorphic maps. As in the classical non-fibred case, we show that petals are the main ingredient. Nevertheless, one expects the properties of the base rotation number should play an important role in the arrangement of the petals. We exhibit examples where the existence and the number of petals depend not just on the complex coordinate of the map, but on the base rotation number. Furthermore, under additional hypothesis on the arithmetics and smoothness of the map, we present a theorem that allows to characterize the local dynamics around a parabolic invariant curve.

Hyperbolization of cocycles by isometries of the Euclidean space

We study hyperbolized versions of cohomological equations that appear with cocycles by isometries of the euclidean space. These (hyperbolized versions of) equations have a unique continuous solution. We concentrate in to know whether or not these solutions converge to a genuine solution to the original equation, and in what sense we can use them as good approximative solutions. The main advantage of considering solutions to hyperbolized cohomological equations is that they can be easily described, since they are global attractors of a naturally defined skew-product dynamics. We also include some technical results about twisted Birkhoff sums and exponential averaging.

Fibred quadratic polynomials can admit two attracting invariant curves

We present an example of a fibred quadratic polynomial admitting two attracting invariant curves. This phenomena cannot occur in the non-fibred setting.