Dynamics in the moduli space of Abelian differentials.
Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory
We continue our study of the dynamics of mappings with small topological degree on projective complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic “equilibrium” measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points...
Dynamics of one-resonant biholomorphisms
Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in whose differentials have one-dimensional family of resonances in the first eigenvalues, (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.
Dynamics of symmetric holomorphic maps on projective spaces.
We consider complex dynamics of a critically finite holomorphic map from Pk to Pk, which has symmetries associated with the symmetric group Sk+2 acting on Pk, for each k ≥1. The Fatou set of each map of this family consists of attractive basins of superattracting points. Each map of this family satisfies Axiom A.
Dynamics on Thurston's sphere of projective measured foliations.
Dynamics semi-conjugated to a subshift for some polynomial mappings in C2.
We study the dynamics near infinity of polynomial mappings f in C2. We assume that f has indeterminacy points and is non constant on the line at infinity L∞. If L∞ is f-attracting, we decompose the Green current along itineraries defined by the indeterminacy points and their preimages. The symbolic dynamics that arises is a subshift on an infinite alphabet.
E. E. Levi convexity and the Hans Lewy problem. Part I : reduction to vanishing theorems
E. E. Levi convexity and the Hans Lewy problem. Part II : vanishing theorems
Each homotopically homogeneous tube in has an affine-homogeneous base.
Each Non-Zero Convolution Operator on the Entire Functions Admits a Continuous Linear Right Inverse.
Earthquakes are analytic.
Éclatement des coefficients des séries entières et deux théorème des Gabrielov.
Éclatements quasi homogènes
Edge-of-the-wedge theorem for elliptic systems
Editorial statement on the paper: The Corona theorem for the ball and polydisc.
Effective formulas for complex geodesics in generalized pseudoellipsoids with applications
We introduce a class of generalized pseudoellipsoids and we get formulas for their complex geodesics in the convex case. Using these formulas we get a description of automorphisms of the pseudoellipsoids. We also solve the problem of biholomorphic equivalence of convex complex ellipsoids without any sophisticated machinery.
Effective formulas for invariant functions - case of elementary Reinhardt domains
We find effective formulas for the invariant functions, appearing in the theory of several complex variables, of the elementary Reinhardt domains. This gives us the first example of a large family of domains for which the functions are calculated explicitly.
Effective Formulas for the carathéodory distance.
Effective freeness and point separation for adjoint bundles.
Effective local finite generation of multiplier ideal sheaves
Let be a psh function on a bounded pseudoconvex open set , and let be the associated multiplier ideal sheaves, . Motivated by global geometric issues, we establish an effective version of the coherence property of as . Namely, given any , we estimate the asymptotic growth rate in of the number of generators of over , as well as the growth of the coefficients of sections in with respect to finitely many generators globally defined on . Our approach relies on proving asymptotic integral...