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Dynamics of certain nonconformal degree-two maps of the plane. (English) Zbl 0816.30015
The authors of the paper under review study the dynamics of the family of rational maps $$(f_{a,c})$$ defined by $z\to | z|^{2a- 2} z^ 2+ c,$ where $$z,c\in \mathbb{C}$$, and $$a$$ is a fixed real number, $$a> 1/2$$. For $$a= 1$$, the family is the well-known family of quadratic polynomials. In their investigation the authors point out the differences in the behaviour of the dynamical systems corresponding to $$a\neq 1$$, respectively to $$a= 1$$. $$f_{1,c}$$ is a conformal map, while for $$a\neq 1$$, $$f_{a,c}$$ is only quasiconformal. One associates to the family under study the filled-in Julia set $$K(a, c)$$, the Julia set $$J(a, c)$$ and the connectedness locus defined as the set: $C_ a= \{c\in \mathbb{C}\mid K(a, c)\text{ is connected}\}.$ Obviously $$C_ 1$$ is the known Mandelbrot set. It is proved that $$C_{1/2}$$ is a union of half- lines, containing the origin, and as $$a\to \infty$$, $$C_ a$$ converges in the Hausdorff topology to the unit disk.
In the holomorphic case there are only two smooth Julia sets and disconnected filled-in Julia sets are totally disconnected. When $$a< 1$$ the Julia set is smooth for an open set of values of $$c$$, and the system may have a periodic attractor, but the critical point is not attracted to it. Structural stable properties for a fixed $$a$$ and $$c$$ close to zero are also investigated. It is analyzed the fixed point structure and the bifurcation occurring when $$a$$ is varied. Finally, a conjecture concerning the connectedness of the set $$C_ a$$ is stated, and some remarks on the topology of this set are done.

##### MSC:
 30C62 Quasiconformal mappings in the complex plane 37B99 Topological dynamics 28A80 Fractals
##### Keywords:
filled-in Julia set; Mandelbrot set; connectedness locus
Mathematica
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