Normalization of bundle holomorphic contractions and applications to dynamics

We establish a Poincar\'e-Dulac theorem for sequences (G_n)_n of holomorphic contractions whose differentials d_0 G_n split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps. Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of CP(k). In this context, our normalization result allows to precisely estimate the distortions of ellipsoids along typical orbits. As an application, we show how the Lyapunov exponents of the equilibrium measure are approximated in terms of the multipliers of the repulsive cycles.Comment: 29 pages, references added, to appear in Ann. Inst. Fourie

Note and calculations concerning elastic dilatancy in 2D glass-glass liquid foams

When deformed, liquid foams tend to raise their liquid contents like immersed granular materials, a phenomenon called dilatancy. We have aready described a geometrical interpretation of elastic dilatancy in 3D foams and in very dry foams squeezed between two solid plates (2D GG foams). Here, we complement this work in the regime of less dry 2D GG foams. In particular, we highlight the relatively strong dilatancy effects expected in the regime where we have predicted rapid Plateau border variations.Comment: 12 pages, 3 tables, 5 figure

The dynamics near quasi-parabolic fixed points of holomorphic diffeomorphisms in C-2

Let F be a germ of holomorphic diffeomorphism of C-2 fixing O and such that dF(O) has eigenvalues 1 and e(itheta) with \e(itheta)\ = 1 and e(itheta) not equal 1. Introducing suitable normal forms for F we define an invariant, nu(F) greater than or equal to 2, and a generic condition, that of being dynamically separating. In the case F is dynamically separating, we prove that there exist nu(F) - 1 parabolic curves for F at O tangent to the eigenspace of 1

Local triple derivations on real C*-algebras and JB*-triples

We study when a local triple derivation on a real JB*-triple is a triple derivation. We find an example of a (real linear) local triple derivation on a rank-one Cartan factor of type I which is not a triple derivation. On the other hand, we find sufficient conditions on a real JB*-triple E to guarantee that every local triple derivation on E is a triple derivation