From Newton's method to exotic basins. II: Bifurcation of the Mandelbrot-like sets (Q2717683)

This article is a continuation of the work of the author in [Fundam. Math. 158, 249-288 (1998; Zbl 1014.37033)]. He studies the family \(\mathcal F = \{f_{a,b}\}\), where NEWLINE\[NEWLINE f_{a,b}(z) = az^2{bz+1-2b \over (2-b)z-1}, \quad a \in \mathbb C \setminus \{0\}, \quad b \in \mathbb C \setminus \{0,1\}. NEWLINE\]NEWLINE This family consists of cubic rational maps with two superattracting fixed points \(0\) and \(\infty\) and a critical point at \(1\). Now, consider the family \(\mathcal N = \{N_\sigma\}\) of Newton maps, where NEWLINE\[NEWLINE N_\sigma=f_{a,b} \quad \text{with} \quad a=\left({\sigma-2 \over 2\sigma-1}\right)^2, \quad b=\sigma. NEWLINE\]NEWLINE It is known that in the parameter plane of \(\mathcal N\) there exist Mandelbrot-like sets (quasiconformally homeomorphic to the standard Mandelbrot set) corresponding to certain sets of parameters \(\sigma\) for which \(N_\sigma^k|_{U_\sigma}\) is quadratic-like on some simply connected domain \(U_\sigma\) with some \(k \geq 2\). NEWLINENEWLINENEWLINEGiven such a Mandelbrot-like set \(M_N\), the author proves the existence of a continuous path of Mandelbrot-like sets joining \(M_N\) via \(M_t\), \(t \in (0,1)\), contained in the family of maps with an attracting fixed point of multiplier \(t\) to a parabolic \(M_1\) contained in the family of maps with a fixed point of multiplier \(1\). Then this path bifurcates into two paths of Mandelbrot-like sets, contained in the set of maps with exotic or non-exotic basins. Here an exotic basin is a multiply connected completely invariant basin of an attracting fixed point, containing \(k\) critical points counted with multiplicity, with \(k\) less than the degree of the map. Moreover, the non-exotic path can be extended in such a way that it terminates with a Mandelbrot-like set in the family of maps conformally conjugate to cubic polynomials with a superattracting fixed point.