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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. 

Abstract We initiate the study of Schrödinger operators with ergodic potentials defined over circle map dynamics, in particular over circle diffeomorphisms. For analytic circle diffeomorphisms and a set of rotation numbers satisfying Yoccoz’s $${{\mathcal{H}}}$$ arithmetic condition, we discuss an extension of Avila’s global theory. We also give an abstract version and a short proof of a sharp Gordon-type theorem on the absence of eigenvalues for general potentials with repetitions. Coupled with the dynamical analysis, we obtain that, for every $$C^{1+BV}$$ circle diffeomorphism, with a super Liouville rotation number and an invariant measure $$\mu $$, and for $$\mu $$-almost all $$x\in{{\mathbb{T}}}^1$$, the corresponding Schrödinger operator has purely continuous spectrum for every Hölder continuous potential $$V$$.