Proofs are given of two theorems of Berezin and Karpelevič, which as far as we know never have been proved correctly. By using eigenfunctions of the Laplace-Beltrami operator it is shown that the spherical functions on a complex Grassmann manifold are given by a determinant of certain hypergeometric functions. By application of this result, it is proved that a certain system of operators, fow which explicit expressions are given, generates the algebra of radial parts of invariant differential operators.

We study the dynamics of polynomial automorphisms of$C$^{$k$}. To an algebraically stable automorphism we associate positive closed currents which are invariant under$f$, considering$f$as a rational map on$P$^{$k$}. These currents give information on the dynamics and allow us to construct a canonical invariant measure which is shown to be mixing.

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The problem of inversion of the attenuated X-ray transformation is solved by an explicit formula. Several subsequent results are also given.