This paper studies the dynamics of the family λ sin z for some values of λ. First we give a description of the Fatou set for values of λ inside the unit disc. Then for values of λ on the unit circle of parabolic type (λ = exp (i2πθ), θ = p/q, (p, q) = 1), we prove that if q is even, there is one q-cycle of Fatou components, if q is odd, there are two q cycles of Fatou components. Moreover the Fatou components of such cycles are bounded. For λ as above there exists a component Dq tangent to the unit disc at λ of a hyperbolic component. There are examples for λ such that the Julia set is the whole complex plane. Finally, we discuss the connectedness locus and the existence of buried components for the Julia set.
There is a classical extension, of Möbius automorphisms of the Riemann sphere into isometries of the hyperbolic space H 3 , which is called the Poincaré extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of H 3 exploiting the fact that any holomorphic covering between Riemann surfaces is Möbius for a suitable choice of coordinates. We show that these extensions define conformally natural homomorphisms on suitable subsemigroups of the semigroup of Blaschke maps. We extend the complex multiplication to a product in H 3 that allows to construct a visual extension of any given rational map.
For À a lattice in PU(2,1), the isometries of the 2-complex ball naturally extend to an action in CP 2 : we prove that the limit set of this action coincides with the complement of the ball in the projective space. We prove that the action is ergodic with respect to a certain measure. Also, we study other kinds of actions: for instance we construct an action whose limit set is the entire CP 2 , but is not minimal although it is algebraically mixing.
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