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

This note supplies examples for the manuscript: Rigidity on the Two-Torus and Sarnak's Conjecture. For every $0<δ<\tfrac12$, we give two constructions of semi-irrational $C^\infty$ diffeomorphisms of $\mathbb{T}^2$ that satisfy the hypotheses of both Theorems 1 and 2 of that manuscript but do not have bounded mean motion, together with totally irrational counterparts satisfying Theorem 1. The first construction is an explicit Anosov-Katok limit, and the second is the time-one map of a smooth special flow followed by Moser normalization. In both cases the map preserves Lebesgue area, its rotation set is a singleton, and its lifted displacement is uniformly $O(n^δ)$ but unbounded. The semi-irrational version has rotation set $\{(α,0)\}$ and satisfies both Theorems 1 and 2. Totally irrational versions have rotation sets $\{(α,α^2)\}$ and $\{(α^2,α)\}$ and satisfy Theorem 1 (Theorem 2 is, by definition, restricted to the semi-irrational case). We also show that there are exactly continuum many examples of each type, including continuum many topological conjugacy classes.