  In this paper, we study the attracting basins of the origin in C^(k+1) for
the polynomial lifts of Lattes examples. We show that their boundaries are
obtained as quotient of a spherical hypersurface and we explicit the
singularities that appear. These domains are surprising, because they are very
close to the ball, and admit a non injective self proper holomorphic map. For
the proof, we desingularize these boundaries into a spherical hypersurface in a
line bundle over a complex torus by using theta fonctions. Finally, we discuss
some examples in dimension 2. We show that some critically finite endomorphisms
of CP^2 considered by Ueda are actually Lattes examples although the one
studied by Fornaess and Sibony is not.
