The full delocalization of eigenstates for the quantized cat map

Abstract

We consider the quantum cat map - a toy model of a quantized chaotic system. We show that its eigenstates are fully delocalized on $\mathbb{T}^2$ in the semiclassical limit (or equivalently that each semiclassical measure is fully supported on $\mathbb{T}^2$). We adapt the proof of a similar result proved for the eigenstates of $-\Delta_g$ on compact hyperbolic surfaces from [arXiv:1705.05019], relying on the fractal uncertainty principle in [arXiv:1612.09040].