On a Slice of the Cubic 2-adic Mandelbrot Set

TL;DR

The paper investigates a cubic $2$-adic slice of the Mandelbrot set in the family $f_t(z) = -\tfrac{3}{2} t (-2z^3+3z^2) + 1$ over $\mathbb{C}_2$, focusing on which parameters yield post-critically bounded dynamics and on the boundary structure. It proves that $t=1$ is a boundary point with asymptotic self-similarity, and it identifies self-similar disk patterns in the parameter space near $t=1$ by alternating PCB and non-PCB regions. Through a Newton polygon analysis, it constructs a sequence $t_n \to 1$ with $f_{t_n}$ PCF of period $n$ and shows nearby disks preserving PCB, revealing a rich, self-similar boundary. Finally, it analyzes the Julia set for $f_1$, uncovering a two-level disk pattern around the repelling fixed point $-\tfrac{1}{2}$ that mirrors the Mandelbrot boundary and echoes Tan Lei’s complex-case observations in the $2$-adic setting.

Abstract

Consider the one-parameter family of cubic polynomials defined by $f_t(z) =-\frac 32 t(-2z^3+3z^2)+1, t \in \mathbb{C}_2$. This family corresponds to a slice of the parameter space of cubic polynomials in $\mathbb{C}_2[z]$. We investigate which parameters in this family belong to the cubic $2$-adic Mandelbrot set, a $p$-adic analog of the classical Mandelbrot set. When $t=1$, $f_t(z)$ is post-critically finite with a strictly preperiodic critical orbit. We establish that this is a non-isolated boundary point on the cubic $2$-adic Mandelbrot set and show asymptotic self-similarity of the Mandelbrot set near this point. Subsequently, we investigate the Julia set for polynomial on the boundary and demonstrate a similarity between the Mandelbrot set at this point and the Julia set, similar to what is seen in the classical complex case.

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Theorems & Definitions (19)

• Definition 1
• Lemma 1
• proof
• Theorem 1
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• Theorem 2
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• Theorem 3
• Remark 1
• Lemma 2