Mathematicians have constructed smooth maps of the two-torus that satisfy a quantitative deviation condition while lacking bounded mean motion, the condition that keeps accumulated motion uniformly controlled around a rotation. The examples also resist being recast, even through nonlinear changes of coordinates, as skew products over circle rotations. The central finding is that meeting the deviation condition does not force either bounded mean motion or a hidden circle-based structure.
Built as a mathematical stress test
This is a theoretical result, not an analysis of observations. The document studies mathematically constructed smooth diffeomorphisms of the two-torus, arranged into two realization methods and two rotation types: semi-irrational and totally irrational. The claims come from deterministic constructions and mathematical estimates rather than an empirical sample.
Two routes to the same separation
The first route is a controlled weakly mixing Anosov–Katok construction. It builds a limit through successive approximations and can be arranged so that the uniform quantitative bound holds for every exponent δ with 0 < δ < 1. Both the semi-irrational and totally irrational versions are available in this branch.
The second route is an explicit weakly mixing special-flow construction using a lacunary Fourier series. In this family, the deviations have an upper bound of order n^δ, meaning they grow no faster than a constant multiple of that power, while still becoming unbounded. The paper says this exponent is sharp for these examples: no smaller deviation exponent can replace it.
The maps keep a single rotation profile
Across all the constructed families, every map preserves Lebesgue area, is weakly mixing with respect to Lebesgue measure, and has a singleton rotation set. That last property means the construction is organized around one rotation vector rather than a collection of possible rotation vectors. The examples therefore retain a tightly specified rotation profile while failing bounded mean motion.
The non-fibred conclusion is equally important. The maps are not topologically conjugate to skew products over circle rotations, including after nonlinear coordinate changes. The result therefore survives a broad change of coordinates rather than depending on one particular description of the maps.
The special-flow examples are explicitly given in both listed rotation types, (α, 0) and (α2, α). The semi-irrational representative uses a strongly super-Liouville α of strong non-Brjuno type, while the totally irrational vector can also be strongly super-Liouville.
A direct test of cited rigidity results
The note also places the examples within the scope of the rigidity results it cites. The semi-irrational maps meet Theorem 1 with Hölder exponent a = 1 and meet Theorem 2 for every finite k ≥ 2. The totally irrational maps meet Theorem 1, but the supplied analysis restricts the Theorem 2 compatibility claim to the semi-irrational case.
On the special-flow side, Moser normalization puts the construction into the stated area-preserving form. The normalized maps retain the singleton rotation set, deviation estimate, failure of bounded mean motion, exponent sharpness and weak mixing. That preservation shows that the main separation survives the area-normalization step.
A broad family within a narrow setting
The result is reported as abundant within the construction scheme. Each construction produces continuum many maps and continuum many topological conjugacy classes of each rotation type, so the phenomenon is not presented as a single exceptional example.
But the scope remains specific. The work concerns constructed smooth area-preserving maps on the two-torus, not an empirical population, and it reports deterministic mathematical conclusions rather than statistical uncertainty. It does not show that every pseudo-rotation satisfying the deviation condition has unbounded mean motion, or that the same conclusions hold outside the smooth area-preserving two-torus setting.
Two boundaries are especially clear in the supplied analysis. Sharpness of the deviation exponent is established for the special-flow examples, not extended there to the Anosov–Katok family. Compatibility with the second cited theorem is limited to the semi-irrational case; the totally irrational examples are covered only by the first theorem.
The document is an arXiv preprint, version 2, dated 27 Aug 2026, and no journal is listed in the supplied metadata.
Paper data and sources
Original title: Examples beyond Bounded Mean Motion for Quantitative Rigidity on the Two-Torus
Authors: Yinshan Chang, Jian Wang, Junchang Zhou
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text