# Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum

### Abstract

We prove almost Lipshitz continuity of spectra of singular quasiperiodic Jacobi matrices and obtain a representation of the critical almost Mathieu family that has a singularity. This allows us to prove that the Hausdorff dimension of its spectrum is not larger than $1/2$ for all irrational frequencies, solving a long-standing problem. Other corollaries include two very elementary proofs of zero measure of the spectrum (e.g., Problem 5 in B. Simon’s list of the 21st century problems) and a similar Hausdorff dimension result for the quantum graph graphene.

## Authors

Svetlana Jitomirskaya

Department of Mathematics, University of California, Irvine, Irvine, CA, USA

Igor Krasovsky

Department of Mathematics, Imperial College London, London, United Kingdom