Anisotropic spaces and nil-automorphisms

TL;DR

This paper develops a new family of anisotropic Banach spaces on Heisenberg nilmanifolds to analyze the spectral properties of partially hyperbolic automorphisms, revealing self-similar spectral structures.

Contribution

It introduces a novel family of anisotropic Banach spaces tailored for nilmanifolds and characterizes the spectral behavior of associated automorphisms, including spectral radius control.

Findings

Essential spectral radius can be made arbitrarily small.

Exterior spectrum matches the spectrum on a specific kernel.

Discrete spectrum exhibits self-similarity and scaling properties.

Abstract

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.