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This paper uses the generalization of the Hukuhara difference for compact convex set to extend the classical notions of Carathéodory differentiability to multifunctions (set-valued maps). Using the Hukuhara difference and affine multifunctions as a local approximation, we introduce the notion of CH-differentiability for multifunctions. Finally, we tackle the study of the relation among the Fréchet differentiability, Hukuhara differentiability, and CH-differentiability. [ABSTRACT FROM AUTHOR]

We introduce a notion stability for subgroups of local complex analytic diffeomorphisms having a common fixed point, in several complex variables. This notion, called L-stability, is inspired in the notion of stability of Lyapunov for singular points and closed orbits of ordinary differential equations. It is also connected to the notion of stability for a proper leaf of a foliation in the classical sense of Reeb. We first classify in terms of the unitary group. Then we prove analytic linearization for a L-stable map and on the classification of L-stable linear groups. This is related to the study of subgroups of U(n), the unitary matrix group. [ABSTRACT FROM AUTHOR]

We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf of a foliation. For dimension one, there is a well-established dictionary relating analytic/formal classification of the group, with its algebraic properties (finiteness, commutativity, solvability, ...). Such system of equivalences also characterizes the existence of suitable integrating factors, i.e., invariant vector fields and one-forms associated to the group. In this paper we search the basic lines of such dictionary for the case of several complex variables groups. For abelian, metabelian, solvable or nilpotent groups we investigate the existence of suitable formal vector fields and closed diferential forms which exhibit an invariance property under the group action. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimension. [ABSTRACT FROM AUTHOR]