A STUDY OF THE DYNAMICS OF λ sin z

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.