Leandro Arosio is currently working on three research projects. The first one, a joint work with F. Bracci, is the study of holomorphic dynamics in the unit ball of C^n. In particular, he is studying commuting self-maps in the unit ball in order to prove simultaneous linearization and to show when a hyperbolic and a parabolic self-maps cannot commute. The main tool here is the canonical Kobayashi hyperbolic semi-model. He is also considering backward dynamics in the ball, trying to adapt Ostapyuk's proof concerning the existence of a backward orbit with bounded step converging to an isolated boundary repelling fixed point to the case of a boundary point z which is not isolated.
This would give as a consequence the existence of a canonical pre-model at the point z.
The second research project, a joint work with E. Wold, studies the existence and the minimal dimension of totally real embeddings in the complex space of a smooth compact real manifold, such that the polynomial convex hull of the image is strictly larger than the image itself, but contains no analytic disc.
The third research project, a joint work with F. Larusson, studies the interactions between Oka theory and dynamics. In particular, he is trying to obtain a dynamical characterization for ellipticity in the case of Stein manifolds.