Alberto Saracco (Università di Parma)

Abstract: While the dynamical behaviour of the iteration of holomorphic functions in one variable is well known, the situation is drastically different in several variables. This should not be a surprise. After all, even from the geometrical point of view the two situations are drastically different: in several variables there is no theorem similar to the Riemann uniformization theorem, and even simple domains as the ball and the polydisk are not biholomorphically equivalent; a holomorphic function of several variables is not determined if known on a set with an accumulation point; there are open domains which are not the maximal natural domain of any holomorphic function (Hartogs' phenomenon). Thus, understanding the dynamical behaviour of the iterations of holomorphic maps, even of automorphisms of C^2, is quite difficult. There are some classes of functions, which can be thought of as being of dimension 1.5, for which it is easier to find results, using theorems of the 1-dimensional theory together with some tools of geometrical flavour. Among these, are the Hénon maps: F(z,w)=(f(z)-\delta w , z) where f is a one-dimensional entire function, and \delta is a complex number. If f is a polynomial, they are a valid playground to understand the behaviour of all polynomial automorphisms of C^2. If f is trascendental, they are not enough to grasp all the possible dynamical behaviours of automorphisms of C^2, but nevertheless they are a starting point. In the first part of the seminar I will present the state of the art of holomorphic dynamic in C^2, while in the second part I will talk about recent results on trascendental Hénon maps, in collaboration with Anna Miriam Benini, Veronica Beltrami and Michela Zedda.