On a Slice of the Cubic 2-Adic Mandelbrot Set

Abstract Consider the one-parameter family of cubic polynomials defined by $$f_t(z) =-\frac{3}{2} t(-2z^3+3z^2)+1, t \in \mathbb {C}_2$$ f t ( z ) = - 3 2 t ( - 2 z 3 + 3 z 2 ) + 1 , t ∈ C 2 . This family corresponds to a slice of the parameter space of cubic polynomials in $$\mathbb {C}_2[z]$$ 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 $$t=1$$ t = 1 , $$f_t(z)$$ 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.