On the dynamics of the singularly perturbed rational maps (Q5962263)

The author studies the dynamics of the family of rational maps \(f_\lambda(z)=z^n+\lambda/z^m\), \(n\geq2\), \(m\geq1\), \(\lambda\in\mathbb C\). Let \(J(f_\lambda)\) be the Julia set of \(f_\lambda\), that is, the set of points in \(\bar{\mathbb C}\) at which the family of iterates of \(f_\lambda\) fails to be a normal family in the sense of Montel. A Sierpinski curve is a planar set that is homeomorphic to the Sierpinski carpet fractal. The Sierpinski carpet fractal is obtained from the unit square by subdividing into nine subsquares, removing the interior of the middle subsquare, then removing the middle ninth of each of the remaining eight subsquares, and continuing inductively. The existence of Sierpinski curves as Julia sets is known from the work of \textit{J. Milnor} and \textit{Tan Lei} [\textit{J. Milnor,} Geometry and dynamics of quadratic rational maps (with an appendix by J. Milnor and Tan Lei). Exp. Math. 2, No. 1, 37--83 (1993; Zbl 0922.58062)]. In 2005, \textit{R. L. Devaney} and \textit{D. M. Look} [Discrete Contin. Dyn. Syst. 13, No. 4, 1035--1046 (2005; Zbl 1106.37031)] give an example which shows the existence of a new type of Sierpinski curve Julia sets, named buried Sierpinski curve Julia sets, for the rational map \(z\mapsto z^{2n}+\lambda/z^{2m+1}\), where \(\lambda\in \mathbb R^-\), \(n\geq1\), \(m\geq0\), and every finite critical point lies in the superattracting cycle. A Sierpinski curve Julia set is called buried if the Fatou domains consist of open sets that are basins of \(\infty\) as well as basins of attracting cycles. In this paper, the author constructs an example of a buried Sierpinski curve Julia set somewhat different from the example given by Devaney and Look. In his example, \(F_\lambda\) has an attracting cycle along the real axis, and the number of the cyclic points is even in \(\mathbb R^-\) while odd in \(\mathbb R^+\). Another goal of the paper is to give an estimate for the location of the bifurcation locus \(\text{BIF\,}(f_\lambda)\) (defined by \textit{C. T. McMullen} in [The Mandelbrot set is universal. Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press (ISBN 0-521-77476-4/pbk). Lond. Math. Soc. Lect. Note Ser. 274, 1--17 (2000; Zbl 1062.37042)]) of the family of maps \(f_\lambda(z)=z^n+\lambda/z^m\), \(n\geq2\), \(m\geq1\), \(\lambda\in\mathbb C\). The authors' result is the following. Theorem 2. The bifurcation locus \(\text{BIF}(f_\lambda)\) is contained in the annular region \[ \Big\{\lambda\;\Big|\;C<|\lambda|<\frac{n}{m}\Big\}, \] where \[ C=\min\left\{\left(\frac19\right)^{1/(m-2)} \left(\frac n{4m}\right)^{m/(m-2)}, \left(\frac {m^{m+1}}{3n(2n)^m}\right)^{2/(n-2)}, \left(\frac{m^m3^{2n}}{n^m}\right)^{1/(2n-m)}\right\}\,. \]