Regularity, quantitative deviation, and non-rigidity of a lacunary skew product

Abstract

Let $\alpha$ be irrational and let $q_j$ be the denominators of its continued-fraction convergents. We study the function \[ h(x)=\sum_{j\geq1}\frac{\cos(2\pi q_jx)}{q_j} \] and the skew product \[f(x,y)=(x+\alpha,y+h(x))\quad\mathrm{mod}\quad\mathbb{Z}^2.\] The function $h$ is H\"older continuous of every exponent below one. A Fourier argument shows that $h$ is not Lipschitz. The map $f$ is a toral pseudo-rotation with rotation vector $(\alpha,0)$, but it has neither bounded mean motion nor $C^0$-rigidity. Suppose $\alpha$ satisfies the Diophantine condition $\mathrm{DC}(\tau)$. Then, $f$ has $(C,1-1/\tau)$-deviation when $\tau>1$; and it has $(C_\delta,\delta)$-deviation for every $0<\delta<1$, but not for $\delta=0$ when $\tau=1$.