# Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus

## TL;DR

This paper demonstrates that certain analytic quasi-periodic cocycles on higher dimensional tori with nontrivial Lyapunov spectra cannot be homotoped to cocycles with dominated splitting, revealing topological obstructions in higher dimensions.

## Contribution

It provides explicit examples showing the failure of dominated splitting in higher dimensional tori, challenging previous results valid in the one-dimensional case.

## Key findings

- Existence of cocycles with nontrivial Lyapunov spectrum
- Homotopy classes lack cocycles with dominated splitting
- Counterexamples to previous theorems in higher dimensions

## Abstract

Consider the space of analytic, quasi-periodic cocycles on the higher dimensional torus. We provide examples of cocycles with nontrivial Lyapunov spectrum, whose homotopy classes do not contain any cocycles satisfying the dominated splitting property. This shows that the main result in the recent work "Complex one-frequency cocycles" by A. \'Avila, S. Jitomirskaya and C. Sadel does not hold in the higher dimensional torus setting.