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In this paper, we prove pure point spectrum for a large class of Schrödinger operators over circle maps with conditions on the rotation number going beyond the Diophantine. More specifically, we develop the scheme to obtain pure point spectrum for Schrödinger operators with monotone bi-Lipschitz potentials over orientation-preserving circle homeomorphisms with Diophantine or weakly Liouville rotation number. The localization is uniform when the coupling constant is large enough. 

Proofs of localization for random Schrödinger operators with sufficiently regular distribution of the potential can take advantage of the fractional moment method introduced by Aizenman–Molchanov [Commun. Math. Phys. 157(2), 245–278 (1993)] or use the classical Wegner estimate as part of another method, e.g., the multi-scale analysis introduced by Fröhlich–Spencer [Commun. Math. Phys. 88, 151–184 (1983)] and significantly developed by Klein and his collaborators. When the potential distribution is singular, most proofs rely crucially on exponential estimates of events corresponding to finite truncations of the operator in question; these estimates in some sense substitute for the classical Wegner estimate. We introduce a method to “lift” such estimates, which have been obtained for many stationary models, to certain closely related non-stationary models. As an application, we use this method to derive Anderson localization on the 1D lattice for certain non-stationary potentials along the lines of the non-perturbative approach developed by Jitomirskaya–Zhu [Commun. Math. Physics 370, 311–324 (2019)] in 2019. 

This work is motivated by an article by Wang, Casati, and Prosen[Phys. Rev. E vol. 89, 042918 (2014)] devoted to a study of ergodicityin two-dimensional irrational right-triangular billiards. Numericalresults presented there suggest that these billiards are generally notergodic. However, they become ergodic when the billiard angle is equalto \pi/2 π / 2 times a Liouvillian irrational, morally a class of irrational numberswhich are well approximated by rationals. In particular, Wang etal. study a special integer counter that reflects the irrationalcontribution to the velocity orientation; they conjecture that thiscounter is localized in the generic case, but grows in the Liouvilliancase. We propose a generalization of the Wang-Casati-Prosen counter:this generalization allows to include rational billiards intoconsideration. We show that in the case of a 45°\!\!:\!45°\!\!:\!90° 45 ° : 45 ° : 90 ° billiard, the counter grows indefinitely, consistent with theLiouvillian scenario suggested by Wang et al.