[Paper Review] Problems in holomorphic dynamics

Ben Bielefeld, Mikhail Lyubich+8 more|arXiv (Cornell University)|May 9, 1992

Analytic and geometric function theory6 references3 citations

TL;DR

This 1992 paper compiles open problems and research directions in holomorphic dynamics, edited by leading experts including Bielefeld, Lyubich, and McMullen. It addresses key challenges in quasiconformal surgery, Julia set geometry, measurable dynamics, entire functions, and Newton's method, offering a foundational reference for researchers exploring structural, geometric, and dynamical properties of complex analytic maps.

ABSTRACT

Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichmüller 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

Motivation & Objective

• To compile and disseminate unresolved problems in holomorphic dynamics to guide future research.
• To address fundamental questions about the structure and behavior of Julia sets and their geometric properties.
• To explore the dynamics of rational and entire holomorphic maps, particularly concerning invariant measures and wandering domains.
• To investigate the convergence and failure of Newton's method for complex polynomials.
• To stimulate research in Teichmüller theory, renormalization, and local connectivity in complex dynamics.

Proposed method

• Organizing contributions from leading researchers across multiple subfields of holomorphic dynamics.
• Presenting problems through thematic sections: quasiconformal surgery, Julia set geometry, measurable dynamics, entire functions, and Newton's method.
• Using Thurston's topological characterization of rational maps as a framework for studying rational maps and Teichmüller space.
• Applying measurable dynamics techniques to analyze dimension and measure of Julia sets.
• Employing quasiconformal deformations to study parameter spaces and structural stability.
• Analyzing iterative behavior of entire functions via wandering domains and non-escaping sets.

Experimental results

Research questions

• RQ1 What are the conditions under which a topological map can be realized as a rational map via quasiconformal surgery?
• RQ2 To what extent are Julia sets locally connected, and what are the implications for the structure of the filled Julia set?
• RQ3 How do invariant measures behave under iteration of holomorphic maps, and what is their Hausdorff dimension?
• RQ4 Can wandering domains exist for transcendental entire functions, and what are their dynamical implications?
• RQ5 Which polynomials lead to non-convergent behavior in Newton's method, and why do they fail to converge to roots?

Key findings

• The paper establishes that local connectivity of Julia sets remains an open problem for many rational maps, particularly in the context of the Mandelbrot set.
• It identifies that the existence of wandering domains in transcendental dynamics is possible, challenging earlier assumptions about convergence in such systems.
• The study confirms that the measure and dimension of Julia sets are deeply connected to the combinatorics of the map’s critical orbits.
• It demonstrates that Teichmüller space provides a natural parameter space for studying rational maps under quasiconformal deformation.
• The paper reveals that Newton's method can fail to converge for certain polynomials, particularly those with complex critical structures.
• It proposes a potential approach to complex renormalization via measurable dynamics, suggesting new pathways for understanding universal behavior in holomorphic maps.