Local connectivity of the Julia set of real polynomials

1995/04/26 by Genadi Levin, Levin, Genadi, Sebastian van Strien +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.math/9504227

openalex publication_date 1995/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

One of the main questions in the field of complex dynamics is the question whether the Mandelbrot set is locally connected, and related to this, for which maps the Julia set is locally connected. In this paper we shall prove the following Main Theorem: Let f be a polynomial of the form f(z)=z^d +c with d an even integer and c real. Then the Julia set of f is either totally disconnected or locally connected. In particular, the Julia set of z²+c is locally connected if c ∈ [-2,1/4] and totally disconnected otherwise.

Cited by

• Polynomial-like dynamics of analytic maps
• Most Fatou and Julia components are small for polynomials

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