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 $-{\mathrm{\Delta }}_g$ on compact hyperbolic surfaces by Dyatlov and Jin (Acta Math. 220:2 (2018), 297–339), relying on the fractal uncertainty principle of Bourgain and Dyatlov (Ann. of Math. $(2)$ 187:3 (2018), 825–867).

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