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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
Software:
Mathematica
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