Narrow Results

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 D_q 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.

For the class ${𝒦}$ of meromorphic functions outside a compact countable set of essential singularities, we study the dynamics of some functions in ${𝒦}$ for which the unit disc and its complement are invariant. These functions are products and compositions of the function $E_1=exp(\frac{z-1}{z+1})$ with Blaschke products, that include the family $E_n=exp\frac{z^n-1}{z^n+1}$, which are the main topic of the article. Slices of the space of parameters of $E_n$ are given with a discussion of their main features. We construct a Poincaré extension of $E_n$ to the hyperbolic three-dimensional space and to the 3-sphere. Also we study their dynamics.