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Consider the one-parameter family of cubic polynomials defined by \(f_t(z) =-\frac{3}{2} t(-2z^3+3z^2)+1, t \in \mathbb {C}_2\) . 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 a 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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On a Slice of the Cubic 2-Adic Mandelbrot Set

  • 会议论文

  • 出版模式：

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  • 发表日期：

    2026年

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  • Jacqueline Anderson,
  • Emerald Stacy,
  • Bella Tobin

摘要

Consider the one-parameter family of cubic polynomials defined by \(f_t(z) =-\frac{3}{2} t(-2z^3+3z^2)+1, t \in \mathbb {C}_2\) . 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 a 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.