Discrete local holomorphic dynamics Informal notes

2007

We investigate the dynamical behaviour of a holomorphic map on an f −invariant subset E of U, where f : U → C k , with U an open, connected and polynomially convex subset of C k. We also prove a Birkhoff type Theorem for holomorphic maps in several complex variables, i.e. given an injective holomorphic map f, defined in a neighborhood of U, with U star-shaped and f (U) a Runge domain, we prove the existence of a forward invariant, maximal, compact and connected subset of U which touches ∂U.

Journal of Mathematical Analysis and Applications, 2009

We investigate the dynamical behaviour of a holomorphic map on a f −invariant subset C of U, where f : U → C k. We study two cases: when U is an open, connected and polynomially convex subset of C k and C ⊂⊂ U, closed in U, and when ∂U has a p.s.h. barrier at each of its points and C is not relatively compact in U. In the second part of the paper, we prove a Birkhoff's type Theorem for holomorphic maps in several complex variables, i.e. given an injective holomorphic map f, defined in a neighborhood of U , with U starshaped and f (U) a Runge domain, we prove the existence of a unique, forward invariant, maximal, compact and connected subset of U which touches ∂U.

2003

Let M be a two-dimensional complex manifold and f : M → M a holomorphic map. Let S ⊂ M be a curve made of fixed points of f , i.e. Fix( f ) = S.W estudy the dynamics near S in case f acts as the identity on the normal bundle of the regular part of S. Besides results

International Mathematics Research Notices, 2013

We prove that the stable manifold of every point in a compact hyperbolic invariant set of a holomorphic automorphism of a complex manifold is biholomorphic to C d , provided that a bunching condition, which is weaker than the classical bunching condition for linearizability, holds.

Eprint Arxiv Math 9205209, 1992

Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichm\"uller space; John Milnor, Thurston's algorithm without critical finiteness; Mary Rees, A possible approach to a complex renormalization problem. 2. Geometry of Julia Sets: Lennart Carleson, Geometry of Julia sets; John Milnor, Problems on local connectivity. 3. Measurable Dynamics: Mikhail Lyubich, Measure and Dimension of Julia Sets; Feliks Przytycki, On invariant measures for iterations of holomorphic maps. 4. Iterates of Entire Functions: Robert Devaney, Open questions in non-rational complex dynamics; Alexandre Eremenko and Mikhail Lyubich, Wandering domains for holomorphic maps. 5. Newton's Method: Scott Sutherland, Bad polynomials for Newton's method

2010

Theorem 0.1: (Identity principle) Let X, Y be two (connected) Riemann surfaces, and f , g ∈ Hol(X,Y ). If the set {z ∈ X | f(z) = g(z)} admits an accumulation point then f ≡ g. Corollary 0.2: Let X, Y be two Riemann surfaces, f ∈ Hol(X,Y ) not constant, and w ∈ Y . Then the set f−1(w) is discrete. Theorem 0.3: (Open mapping theorem) Let X, Y be two Riemann surfaces, and f ∈ Hol(X,Y ) not constant. Then f(X) is open in Y . In particular, f is an open mapping.

2011

We give a sufficient condition for the abstract basin of attraction of a sequence of holomorphic self-maps of balls in \mathbb{C}^{d} to be biholomorphic to \mathbb{C}^{d}. As a consequence, we get a sufficient condition for the stable manifold of a point in a compact hyperbolic invariant subset of a complex manifold to be biholomorphic to a complex Euclidean space. Our result immediately implies previous theorems obtained by Jonsson-Varolin and by Peters; in particular, we prove (without using Oseledec's theory) that the stable manifold of any point where the negative Lyapunov exponents are well-defined is biholomorphic to a complex Euclidean space. Our approach is based on the solution of a linear control problem in spaces of subexponential sequences, and on careful estimates of the norm of hte conjugacy operator by a lower triangular matrix on the space of \textit{k}-homogeneous polynomial endomorphisms of \mathbb{C}^{d}.

American Journal of Mathematics, 2000

We shall describe a canonical procedure to associate to any (germ of) holomorphic self-map F of C n fixing the origin so that dF O is invertible and non-diagonalizable an n-dimensional complex manifold M , a holomorphic map π: M → C n , a point e ∈ M and a (germ of) holomorphic self-mapF of M such that: π restricted to M \ π −1 (O) is a biholomorphism between M \ π −1 (O) and C n \ {O}; π •F = F • π; and e is a fixed point ofF such that dF e is diagonalizable. Furthermore, we shall use this construction to describe the local dynamics of such an F nearby the origin when sp(dF O ) = {1}.

Applied Mathematics Letters, 2008

One of the interesting areas in the study of the local dynamics in several complex variables is the dynamics near the origin O of maps tangent to the identity, that is of germs of holomorphic self-maps f : n → n such that f (O) = O and d f O = id. When n = 1 the dynamics is described by the known Leau-Fatou flower theorem but when n > 1, we are still far from understanding the complete picture, even though very important results have been obtained in recent years (see, e.g., [2,7,10,19]). In this note we want to investigate conditions ensuring the existence of parabolic curves (the two-variable analogue of the petals in the Leau-Fatou flower theorem) for maps tangent to the identity in dimension 2. Using simple examples, we prove that these conditions are not, generally, sufficient.

The Michigan Mathematical Journal, 2001

Contemporary Mathematics, 2011

Two dynamical systems u t (t, •) = (Lu)(t, •), L = A, B, u(0, •) = f , which can serve as toy models for infinite systems of interacting particles in continuum are studied. Here u(t, •) and f are holomorphic functions in some K ⊂ C, and A, B are linear operators in a certain Banach space E of such functions. It is proven that both A and B generate C 0 semigroups and hence the above Cauchy problems have solutions in E. In some particular cases, ergodicity and reversibility are proven.

Proceedings of the American Mathematical Society

This note presents a method to study center families of periodic orbits of complex holomorphic differential equations near singularities, based on some iteration properties of fixed point indices. As an application of this method, we will prove Needham's theorem in a more general version. Lemma 1. Let f : ∆ n → C n be a holomorphic mapping such that 0 ∈ ∆ n is an isolated fixed point of f. Then µ f (0) ≥ 1, and the equality holds if and only if the Jacobian matrix f ′ (0) of f at 0 has no eigenvalue equal to 1.

The Nepali Mathematical Sciences Report

In this paper, we prove that the escaping set of a transcendental semi group is S-forward invari-ant. We also prove that if a holomorphic semi group is a belian, then the Fatou, Julia, and escaping sets are S-completely invariant. We also investigate certain cases and conditions that the holomorphic semi group dynamics exhibits the similar dynamical behavior just like a classical holomorphic dynamics. Frequently, we also examine certain amount of connections and contrasts between classical holomorphic dynamics and holomorphic semi group dynamics.

Inventiones Mathematicae, 1995