Let (X, ω) be a compact Kähler manifold. As discovered in the late 1980s by Mabuchi, the set H 0 of Kähler forms cohomologous to ω has the natural structure of an infinite dimensional Riemannian manifold. We address the question whether any two points in H 0 can be connected by a smooth geodesic, and show that the answer, in general, is "no". The Monge-Ampère foliation.Let Y be an m + 1 dimensional complex manifold, ω a real (1, 1) form on it, of class C 1 , dω = 0. If rk ω ≡ m, the kernels of ω form an integrable subbundle of T Y , and so Y is foliated by Riemann surfaces, whose tangent spaces are the kernels of ω. The foliation is of class C 1 . If w is a locally defined potential of

Until recently, little was known about the existence of wandering Fatou components for rational maps in more than one complex variables. In 2014, examples of wandering Fatou components were constructed in Astorg et al. [1] for polynomial skew-products with an invariant parabolic fiber. In 2004 Lilov already proved the non-existence of wandering Fatou components for polynomial skew-products in the basin of an invariant super-attracting fiber. In the current work we investigate in how far his methods carry over to the geometrically attracting case. Lilov proved a stronger statement, namely that the orbit of any horizontal disk in the super-attracting basin must eventually intersect a fattened Fatou components. Here we give explicit constructions to show that this stronger result is false in the geometrically attracting case. However, we also prove that the constructed disks do not give rise to wandering Fatou components. Our construction therefore leaves open the existence of wandering Fatou components in the geometrically attracting case, showing however that the situation is significantly more complicated than in the super-attracting case.

Given a germ of biholomorphism F∈prefixDifffalse(Cn,0false)$F\in \operatorname{Diff}(\mathbb {C}^n,0)$ with a formal invariant curve Γ$\Gamma$ such that the multiplier of the restricted formal diffeomorphism F|normalΓ$F|_\Gamma$ is a root of unity or satisfies |false(F|Γfalse)′(0)|<1$|(F|_\Gamma )^{\prime }(0)|<1$, we prove that either Γ$\Gamma$ is contained in the set of periodic points of F$F$ or there exists a finite family of stable manifolds of F$F$ where all the orbits are asymptotic to Γ$\Gamma$ and whose union eventually contains every orbit asymptotic to Γ$\Gamma$. This result generalizes to the case where Γ$\Gamma$ is a formal periodic curve.

Let F : (C 2 , O) → (C 2 , O) be a germ tangent to the identity. Assume F has a characteristic direction [v]. In [Hak] Hakim gives conditions to guarantee the existence of an attracting basin to the origin along [v], in the case of [v] a non-degenerate characteristic direction. In this paper we give conditions to guarantee the existence of basins along [v] in the case of [v] a degenerate characteristic direction.

We study topological properties of attracting sets for automorphisms of C k . Our main result is that a generic volume preserving automorphism has a hyperbolic fixed point with a dense stable manifold. On the other hand, we show that an attracting set can only contain a neighborhood of the fixed point if it is an attracting fixed point. We will see that the latter does not hold in the non-autonomous setting.