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Bounds on the radius of the $$p$$-adic Mandelbrot set. (English) Zbl 1300.11123
The paper studies a $$p$$-adic version of the Mandelbrot set $\mathcal{M}:=\{c\in \mathbb{C}: \mathrm{the critical orbit of } f_c(z)=z^2+c \mathrm{ is bounded} \}$ in the complex dynamics by replacing $$\mathbb{C}$$ with the field $$\mathbb{C}_p$$ of $$p$$-adic complex numbers and $$f_c$$ with higher order polynomials.
Via conjugation by an affine linear transformation, every polynomial of degree $$d$$ defined over $$\mathbb{C}_p$$ can be written as a monic polynomial with $$f(0)=0$$. Let $$\mathcal{P}_{d, p}$$ be the parameter space of such monic polynomials. The $$p$$-adic Mandelbrot set $$\mathcal{M}_{d,p}$$ is the subset of $$\mathcal{P}_{d, p}$$ such that the critical orbits of any polynomial with parameters in $$\mathcal{M}_{d,p}$$ are bounded. The author estimates the critical radius of the $$\mathcal{M}_{d,p}$$ defined as $$r_{d, p}:= \sup_{f\in \mathcal{M}_{d,p}} \max_{c\in \mathbb{C}_p, f'(c)=0} \{-v_p(c)\}.$$ The main theorem shows that $$r(d,p)=p/(d-1)$$ if $$d/2<p<d$$ and $$r(d,p)=0$$ if $$p=d/2$$. An elementary proof is also given for the known result stating that $$r(d,p)=0$$ when $$p>d$$ or $$d=p^k$$. It is remarked that the value of $$r(d,p)$$ can be useful in searching for all post-critically finite polynomials over a given number field.
In the last section, the one-parameter family of cubic polynomials $$f_t(z)=z^3-{3 \over 2}tz^2$$ over $$\mathbb{C}_2$$ is studied. It is shown that the boundary of the $$2$$-adic Mandelbrot set for this family is complicated and fractal-like.

##### MSC:
 11S82 Non-Archimedean dynamical systems 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
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