The range of holomorphic maps at boundary points

We prove a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains. That is, we prove that the local image of a holomorphic map $f:D\to D'$ close to a boundary regular contact point p\in \partial D where the Jacobian is bounded away from zero along normal non-tangential directions has to eventually contain every cone (and more generally every region which is Kobayashi asymptotic to a cone) with vertex at f(p)

DYNAMICS OF TRANSCENDENTAL HENON MAPS

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental maps to consider. In this paper we find inspiration from the extensive work on the dynamics of complex H´enon maps. We introduce the family of transcendental H´enon maps, and study their dynamics, emphasizing the description of Fatou components. We prove that the classification of the recurrent invariant Fatou components is similar to that of polynomial H´enon maps, and we give examples of Baker domains and wandering domains

Squeezing functions and Cantor sets

We construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function. Finally we show that complements of Cantor sets arising as Julia sets of quadratic polynomials have degenerate squeezing functions, despite of having Hausdorff dimension arbitrarily close to two

Dynamics of transcendental Henon maps III: infinite entropy

Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental Henon maps offers the potential of combining ideas from transcendental dynamics in one variable and the dynamics of polynomial Henon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two