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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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