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Abelian integrals in holomorphic foliations.

Hossein Movasati (2004)

Revista Matemática Iberoamericana

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singularity in the projective space of dimension two. Also we calculate higher Melnikov functions under some...

Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps

Feliks Przytycki (1994)

Fundamenta Mathematicae

We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if A is completely invariant (i.e. f - 1 ( A ) = A ), and if μ is an arbitrary f-invariant measure with positive Lyapunov exponents on ∂A, then μ-almost every point q ∈ ∂A is accessible along a curve from A. In fact, we prove the accessibility of every “good” q, i.e. one for which “small neigh bourhoods arrive at large scale” under iteration of f. This generalizes the...

Algebraic degrees for iterates of meromorphic self-maps of Pk.

Viêt-Anh Nguyên (2006)

Publicacions Matemàtiques

We first introduce the class of quasi-algebraically stable meromorphic maps of Pk. This class is strictly larger than that of algebraically stable meromorphic self-maps of Pk. Then we prove that all maps in the new class enjoy a recurrent property. In particular, the algebraic degrees for iterates of these maps can be computed and their first dynamical degrees are always algebraic integers.

An algebraic formulation of Thurston’s characterization of rational functions

Kevin M. Pilgrim (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston’s characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy classes of simple closed curves induced by a rational function. The resulting finiteness results yield new information on the global dynamics of the pullback map on Teichmüller space used in the proof of the characterization theorem.

Analytic invariants for the 1 : - 1 resonance

José Pedro Gaivão (2013)

Annales de l’institut Fourier

Associated to analytic Hamiltonian vector fields in ℂ 4 having an equilibrium point satisfying a non semisimple 1 : - 1 resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.

Baker domains for Newton’s method

Walter Bergweiler, David Drasin, James K. Langley (2007)

Annales de l’institut Fourier

For an entire function f let N ( z ) = z - f ( z ) / f ′ ( z ) be the Newton function associated to f . Each zero ξ of f is an attractive fixed point of N and is contained in an invariant component of the Fatou set of the meromorphic function N in which the iterates of N converge to ξ . If f has an asymptotic representation f ( z ) ∼ exp ( - z n ) , n ∈ ℕ , in a sector | arg z | < ε , then there exists an invariant component of the Fatou set where the iterates of N tend to infinity. Such a component is called an invariant Baker domain.A question in the opposite direction...

Borel summation and splitting of separatrices for the Hénon map

Vassili Gelfreich, David Sauzin (2001)

Annales de l’institut Fourier

We study two complex invariant manifolds associated with the parabolic fixed point of the area-preserving Hénon map. A single formal power series corresponds to both of them. The Borel transform of the formal series defines an analytic germ. We explore the Riemann surface and singularities of its analytic continuation. In particular we give a complete description of the “first” singularity and prove that a constant, which describes the splitting of the invariant manifolds, does not vanish. An interpretation...

Bounded geometry of quadrilaterals and variation of multipliers for rational maps

Kevin M. Pilgrim (2004)

Fundamenta Mathematicae

Let Q be the unit square in the plane and h: Q → h(Q) a quasiconformal map. When h is conformal off a certain self-similar set, the modulus of h(Q) is bounded independent of h. We apply this observation to give explicit estimates for the variation of multipliers of repelling fixed points under a "spinning" quasiconformal deformation of a particular cubic polynomial.

Currently displaying 21 – 40 of 378