We construct canonical intertwining semi-models with Kobayashi hyperbolic base space for holomorphic self-maps of complex manifolds which are univalent on some absorbing cocompact hyperbolic domain. In particular, in the unit ball we solve the Valiron equation for hyperbolic univalent self-maps and for hyperbolic semigroups.2000 Mathematics Subject Classification. Primary 32H50; Secondary 39B12, 26A18.

We present a new geometric construction of Loewner chains in one and several complex variables which holds on complete hyperbolic complex manifolds and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence, we obtain a univalent solution (f (t) : M -> N) of any Loewner-Kufarev PDE. The problem of finding solutions given by univalent mappings (f (t) : M -> a", (n) ) is reduced to investigating whether the complex manifold a(a) (ta parts per thousand yen0) f (t) (M) is biholomorphic to a domain in a", (n) . We apply such results to the study of univalent mappings f: B (n) -> a", (n)

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 Communicated by Ngaiming Mok.

We prove that given a Herglotz vector field on the unit ball of C^n of the form H(z,t)=(a_1 z_1,\dots,a_n z_n)+O(|z|^2) with Re a_j<0 for all j, its evolution family admits an associated Loewner chain, which is normal if no real resonances occur. Hence the Loewner-Kufarev PDE admits a solution defined for all positive times

Abstract. We characterize infinitesimal generators on complete hyperbolic complex manifolds without any regularity assumption on the Kobayashi distance. This allows to prove a general Loewner type equation with regularity of any order d ∈ [1, +∞]. Finally, based on these results, we focus on some open problems naturally arising.