Sierpinski-curve Julia sets and singular perturbations of complex polynomials. (English) Zbl 1098.37043
Summary: We consider the family of rational maps of the complex plane given by
\[
z^2+\frac{\lambda}{z^2}
\]
where \(\lambda\) is a complex parameter. We regard this family as a singular perturbation of the simple function \(z^2\). We show that, in any neighborhood of the origin in the parameter plane, there are infinitely many open sets of parameters for which the Julia sets of the corresponding maps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However, we also show that parameters corresponding to different open sets have dynamics that are not conjugate.
MSC:
37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |
30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |