A study of the dynamics of lambda sin z

Abstract

This paper studies the dynamics of the family lambda sin z for some values of lambda. First we give a description of the Fatou set for values of lambda inside the unit disc. Then for values of lambda on the unit circle of parabolic type (lambda = exp(i2pitheta), theta = 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 lambda as above there exists a component D-q. tangent to the unit disc at lambda of a hyperbolic component. There are examples for lambda 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.