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arXiv · 2211.12537

On Dynamical Parameter Space of Cubic Polynomials with a Parabolic Fixed Point

Runze Zhang
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Abstract

This article focus on the connected locus of the cubic polynomial slice $Per_1(\lambda)$ with a parabolic fixed point of multiplier $\lambda=e^{2\pi i\frac{p}{q}}$. We first show that any parabolic component, which is a parallel notion of hyperbolic component, is a Jordan domain. Moreover, a continuum $\mathcal{K}_\lambda$ called the central part in the connected locus is defined. This is the natural analogue to the closure of the main hyperbolic component of $Per_1(0)$. We prove that $\mathcal{K}_\lambda$ is almost a double covering of the filled-in Julia set of the quadratic polynomial $P_{\lambda}(z) = \lambda z+z^2$.

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Runze Zhang. 2022-11-22. On Dynamical Parameter Space of Cubic Polynomials with a Parabolic Fixed Point. https://arxiv.org/abs/2211.12537

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