Examples beyond Bounded Mean Motion for Quantitative Rigidity on the Two-Torus

Abstract

This note supplies genuinely non-fibred examples for the manuscript: Rigidity on the Two-Torus and Sarnak's Conjecture. For every $0<\delta <\frac{1}{2}$, we construct $C^{\infty }$ Lebesgue-area-preserving pseudo-rotations of ${\mathbb{T}}^2$ which satisfy the $(C,\delta )$-deviation condition but do not have bounded mean motion. We give both semi-irrational and totally irrational rotation vectors and two realizations: a controlled weakly mixing Anosov--Katok construction and an explicit weakly mixing special flow construction. Weak mixing is used as a conjugacy-invariant obstruction to every continuous circle-rotation factor. Consequently, none of the resulting maps is topologically conjugate, by a linear or nonlinear change of coordinates, to a skew product over a circle rotation. In the special-flow realization, the same lacunary Fourier series simultaneously gives weak mixing, the sharp upper bound $O(n^{\delta })$, and unbounded deviations; in fact no smaller deviation exponent is possible. The semi-irrational examples meet the assumptions of Theorems~1 and~2 of the cited manuscript, whereas the totally irrational examples meet those of Theorem~1. Each construction produces continuum many maps and continuum many topological conjugacy classes of each rotation type.

ChatGPT was used to assist in generating concrete examples following instructions on the use of the Anosov--Katok method and the construction of special flows. All mathematical content was verified by the authors