Spectral Theory of Schrödinger Operators over Circle Diffeomorphisms

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 $V$.

Hausdorff dimension of invariant measure of circle diffeomorphisms with a break point

We prove that, for almost all irrational $\unicode[STIX]{x1D70C}\in (0,1)$, the Hausdorff dimension of the invariant measure of a $C^{2+\unicode[STIX]{x1D6FC}}$-smooth $(\unicode[STIX]{x1D6FC}\in (0,1))$ circle diffeomorphism with a break of size $c\in \mathbb{R}_{+}\backslash \{1\}$, with rotation number $\unicode[STIX]{x1D70C}$, is zero. This result cannot be extended to all irrational rotation numbers.

A circle diffeomorphism with breaks that is absolutely continuously linearizable

In this paper we answer positively to a question of whether it is possible for a circle diffeomorphism with breaks to be smoothly conjugate to a rigid rotation in the case where its breaks are lying on pairwise distinct trajectories. An example constructed is a piecewise linear circle homeomorphism that has four break points lying on distinct trajectories and whose invariant measure is absolutely continuous with respect to the Lebesgue measure. The irrational rotation number for our example can be chosen to be a Roth number, but not of bounded type.

Dimensional characteristics of invariant measures for circle diffeomorphisms

We consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number τ there exists a C∞ circle diffeomorphism with rotation number τ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value 0≤β≤1.

Topological conjugacy of circle diffeomorphisms

The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by Denjoy [1]. In [5] one of us gave a new criterion. There is an example satisfying Denjoy's bounded variation condition rather than the Zygmund condition of [5], and vice versa. This paper will give the third criterion which is implied by either of the above criteria.