# Semiclassical measures on hyperbolic surfaces have full support

## TL;DR

This paper proves that all semiclassical measures derived from Laplacian eigenfunctions on compact hyperbolic surfaces are fully supported on the entire cosphere bundle, utilizing the fractal uncertainty principle.

## Contribution

It introduces the application of the fractal uncertainty principle to establish full support of semiclassical measures on hyperbolic surfaces.

## Key findings

- Semiclassical measures have full support on the cosphere bundle.
- The fractal uncertainty principle is effective in this geometric setting.
- Supports the quantum unique ergodicity conjecture in this context.

## Abstract

We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated in [arXiv:1504.06589] and proved for porous sets in [arXiv:1612.09040].