Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces.
Generalized Staircases: Recurrence and Symmetry
We study infinite translation surfaces which are -covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.
Generalized transversely projective structure on a transversely holomorphic foliation.
Geography of the cubic connectedness locus : intertwining surgery
Geometry of currents, intersection theory and dynamics of horizontal-like maps
We introduce a geometry on the cone of positive closed currents of bidegree and apply it to define the intersection of such currents. We also construct and study the Green currents and the equilibrium measure for horizontal-like mappings. The Green currents satisfy some extremality properties. The equilibrium measure is invariant, mixing and has maximal entropy. It is equal to the intersection of the Green currents associated to the horizontal-like map and to its inverse.
Hausdorff convergence and the limit shape of the unicorns.
Hausdorff dimension and measures on Julia sets of some meromorphic maps
We study the Julia sets for some periodic meromorphic maps, namely the maps of the form where h is a rational function or, equivalently, the maps . When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller than 1....
High-order phase transitions in the quadratic family
We give the first example of a transitive quadratic map whose real and complex geometric pressure functions have a high-order phase transition. In fact, we show that this phase transition resembles a Kosterlitz-Thouless singularity: Near the critical parameter the geometric pressure function behaves as near , before becoming linear. This quadratic map has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.
Holomorphic flows and complex structures on products of odd dimensional spheres.
Holomorphic foliations by curves on with non-isolated singularities
Let be a holomorphic foliation by curves on . We treat the case where the set consists of disjoint regular curves and some isolated points outside of them. In this situation, using Baum-Bott’s formula and Porteuos’theorem, we determine the number of isolated singularities, counted with multiplicities, in terms of the degree of , the multiplicity of along the curves and the degree and genus of the curves.
Holomorphic solutions to linear first-order functional differential equations.
Hubbard trees
We provide a full classification of postcritically finite polynomials as dynamical systems by means of Hubbard trees. The information encoded in these objects is solid enough to allow us recover all the relevant statical and dynamical aspects of the corresponding Julia sets.
Hyperbolic components of polynomials with a fixed critical point of maximal order
Hyperbolic components of the complex exponential family
We describe the structure of the hyperbolic components of the parameter plane of the complex exponential family, as started in [1]. More precisely, we label each component with a parameter plane kneading sequence, and we prove the existence of a hyperbolic component for any given such sequence. We also compare these sequences with the more commonly used dynamical kneading sequences.
Hyperbolic measure of maximal entropy for generic rational maps of
Let be a dominant rational map of such that there exists with for all . Under mild hypotheses, we show that, for outside a pluripolar set of , the map admits a hyperbolic measure of maximal entropy with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of to . This provides many examples where non uniform hyperbolic dynamics is established.One of the key tools is to approximate the graph of a meromorphic...
Hyperbolicité des polynômes fibrés
Hyperbolicity of renormalization of critical circle maps
Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive
Hypertranscendency of conjugacies in complex dynamics.
Immediate and Virtual Basins of Newton’s Method for Entire Functions
We investigate the well known Newton method to find roots of entire holomorphic functions. Our main result is that the immediate basin of attraction for every root is simply connected and unbounded. We also introduce “virtual immediate basins” in which the dynamics converges to infinity; we prove that these are simply connected as well.