Anisotropic spaces and nil-automorphisms

We introduce a family of geometric anisotropic Banach spaces on Heisenberg nilmanifolds and study the spectrum of the composition operator associated to partially hyperbolic automorphisms. Choosing amongst the family of Banach spaces, it is possible to make the essential spectral radius arbitrarily small. We show that the exterior part of the discrete spectrum coincides with the spectrum restricted to the kernel of one of the operators associated to the nil-automorphism. Moreover we show that the remainder of the discrete spectrum is self-similar, it is given by scaled copies of the exterior part.