Complex and Real Dynamics for the family λ tan(z)

This article is based on my lecture at the Complex Dynamics Workshop of the Research Institute of Mathematics at Kyoto University in October 2001. It is an exposition of joint work with Janina Kotus. The tangent family fλ(z) = λ tan(z) is the meromorphic analogue of the quadratic family z +c. The functions fλ(z) are characterized by their mapping properties: they are, up to scale, the only meromorphic functions fixing zero with no critical points and two symmetric asymptotic (omitted) values. The classification of stable behavior is essentially the same as for the quadratic family. Again like the quadratic family, the parameter plane has a combinatorial description based on the orbit of the singular value, appropriately interpreted. The real axis plays a special role for the quadratic family. For real values of the parameter, the critical value is real and so is its forward orbit. Studying the orbit of the critical value, we can understand the observed period doubling and renormalization. For the tangent family, the imaginary axis plays a similar role. If the parameter lies on the imaginary axis, the asymptotic values are real as are even of iterates. Restricting our attention to the second iterate f λ for λ = iy ∈ =, we again observe period doubling. In this paper we give an overview of the dynamical theory for the tangent family and describe the period doubling phenomena and some of its consequences. For details and proofs see [1, 3, 4, 5, 6] and the references cited therein.