On a Slice of the Cubic 2-adic Mandelbrot Set

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.