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Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory

Jeffrey Diller, Romain Dujardin, Vincent Guedj (2010)

Annales scientifiques de l'École Normale Supérieure

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

Filippo Bracci, Dmitri Zaitsev (2013)

Journal of the European Mathematical Society

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in C n whose differentials have one-dimensional family of resonances in the first m eigenvalues, m ≤ n (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.

Kohei Ueno (2007)

Publicacions Matemàtiques

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 semi-conjugated to a subshift for some polynomial mappings in C2.

Gabriel Vigny (2007)

Publicacions Matemàtiques

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.

Effective formulas for complex geodesics in generalized pseudoellipsoids with applications

Włodzimierz Zwonek (1995)

Annales Polonici Mathematici

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 local finite generation of multiplier ideal sheaves

Dan Popovici (2010)

Annales de l’institut Fourier

Let ϕ be a psh function on a bounded pseudoconvex open set Ω ⊂ ℂ n , and let ℐ ( m ϕ ) be the associated multiplier ideal sheaves, m ∈ ℕ ☆ . Motivated by global geometric issues, we establish an effective version of the coherence property of ℐ ( m ϕ ) as m → + ∞ . Namely, given any B ⋐ Ω , we estimate the asymptotic growth rate in m of the number of generators of ℐ ( m ϕ ) | B over 𝒪 Ω , as well as the growth of the coefficients of sections in Γ ( B , ℐ ( m ϕ ) ) with respect to finitely many generators globally defined on Ω . Our approach relies on proving asymptotic integral...

Effective Nullstellensatz for arbitrary ideals

János Kollár (1999)

Journal of the European Mathematical Society

Let f i be polynomials in n variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials g i such that ∑ g i f i = 1 . The effective versions of this result bound the degrees of the g i in terms of the degrees of the f j . The aim of this paper is to generalize this to the case when the f i are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.

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