[Paper Review] Discrete holomorphic local dynamical systems

Marco Abate|ArXiv.org|Mar 19, 2009

Mathematical Dynamics and Fractals40 references4 citations

TL;DR

This survey investigates discrete holomorphic local dynamical systems at a fixed point in complex manifolds, focusing on the structure of the stable set and the dynamics within it. It establishes conditions under which such systems are topologically conjugated to simpler normal forms, particularly when the derivative has eigenvalue 1 with multiplicity one and other eigenvalues of modulus less than one, yielding a complete topological classification under non-resonance and Bryuno-type conditions.

ABSTRACT

This is a survey on local dynamics of holomorphic maps in one and several complex variables, discussing in particular normal forms and the structure of local stable sets in the non-hyperbolic case, and including several proofs and a vast bibliography. It updates and enlarge the previous survey arXiv:math/0310089v1.

Motivation & Objective

• To classify discrete holomorphic local dynamical systems at a fixed point in complex manifolds via topological and holomorphic conjugacy.
• To determine when such systems are conjugated to simpler normal forms, especially in the case of eigenvalue 1 with multiplicity one.
• To identify conditions under which invariant curves or manifolds exist, particularly in the presence of eigenvalue 1 and irrational rotation numbers.
• To explore the role of invariants and resonance conditions in classifying dynamics near fixed points.
• To extend known results on linearization and normal forms to semi-attractive and non-linearizable cases.

Proposed method

• Use of local coordinates to express a holomorphic map in a form where the part tangent to the identity is isolated, and the rest is of lower modulus.
• Application of the Pöschel–Bryuno theory to linearize dynamics on invariant disks when the rotation number satisfies the Bryuno condition.
• Employment of normal form theory and results on normally hyperbolic systems to classify dynamics under non-resonance conditions.
• Use of characteristic directions and generalized eigenspaces to construct invariant manifolds tangent to specific directions.
• Application of theorems by Rivi and Di Giuseppe to classify systems with eigenvalue 1 and other eigenvalues of modulus less than one.
• Use of holomorphic invariants to determine topological conjugacy classes, especially in generic cases.

Experimental results

Research questions

• RQ1 Under what conditions is a holomorphic local dynamical system at a fixed point topologically conjugated to its linear part or a simple normal form?
• RQ2 When do invariant complex curves or manifolds exist in the presence of eigenvalue 1 and other eigenvalues of modulus less than one?
• RQ3 What role do resonance conditions and the Bryuno condition play in the linearization of dynamics near a fixed point with eigenvalue 1?
• RQ4 How do holomorphic invariants distinguish between topologically conjugate dynamical systems in the non-linearizable case?
• RQ5 What is the structure of the stable set and how does it relate to the existence of attracting or parabolic behavior?

Key findings

• When the derivative at the fixed point has eigenvalue 1 with algebraic and geometric multiplicity one and all other eigenvalues have modulus less than one, the system is topologically conjugated to either the linear part or a specific non-linear normal form.
• Under non-resonance and Bryuno condition, the dynamics on a 1-dimensional invariant disk is conjugated to an irrational rotation, ensuring linearization on that disk.
• For maps with eigenvalue 1 and non-resonant eigenvalues of modulus less than one, the topological classification yields exactly two possible classes: the linear map or a specific non-linear perturbation.
• The existence of parabolic curves tangent to the eigenspace of eigenvalue 1 is guaranteed under generic conditions involving two new holomorphic invariants, as shown by Bracci and Molino.
• When eigenvalue 1 has multiplicity greater than one, the dynamics can be decomposed into a part tangent to the identity and a contracting part, with invariant manifolds emerging along non-degenerate characteristic directions.
• In dimension two, if the non-1 eigenvalue satisfies the Bryuno condition, a 1-dimensional invariant holomorphic disk exists where the map is conjugated to a rotation.