Anisotropic spaces and nil-automorphisms

TL;DR

This work analyzes transfer operators for partially hyperbolic automorphisms of Heisenberg nilmanifolds by building a tailored family of anisotropic Banach spaces $\mathcal{B}_N^{p,q}$ that capture contraction along $E_V$, expansion along $E_W$, and neutral behaviour along $E_Z$. The operator $\mathcal{L}$ is shown to be quasi-compact on these spaces, with essential spectral radius bounded by $\lambda^{-\min\{p,q\}}$, enabling a precise description of resonances. The peripheral spectrum on $\ker_N(V)$ is fully characterized as a finite set $\{\lambda^{1/2}\mu_j\}$, and a subsequent inductive argument yields the full spectrum as scaled copies of these resonances, i.e., $\{\lambda^{-k}\mu_j\}$. This spectral data facilitates resonance expansions and deviations of ergodic averages, and it lays groundwork for extending the framework to non-algebraic systems via renormalization techniques.

Abstract

We introduce geometric anisotropic Banach spaces on Heisenberg nilmanifolds and study the spectrum of the transfer operator associated to partially hyperbolic automorphisms. For these systems complete spectral data is obtained.

Theorems & Definitions (56)

• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof