An entire transcendental family with a persistent Siegel disc

Abstract

We study the class of entire transcendental maps of finite order with one critical point and one asymptotic value, which has exactly one finite pre-image, and having a persistent Siegel disc. After normalization, this is a one parameter family f a with which includes the semi-standard map at , approaches the exponential map when and a quadratic polynomial when . We investigate the stable components of the parameter plane (capture components and semi-hyperbolic components) and also some topological properties of the Siegel disc in terms of the parameter.