Rigidity on the two-torus and Sarnak's conjecture

Yinshan Chang; Jian Wang; Junchang Zhou

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arXiv · 2608.26976

Rigidity on the two-torus and Sarnak's conjecture

Yinshan Chang·Jian Wang·Junchang Zhou

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Abstract

We establish quantitative rigidity results for pseudo-rotations of the two-torus under a $(C,\delta)$-deviation condition relative to their rotation vectors. The main ingredient is a quantitative free-disk estimate that converts bounds on orbit deviation into explicit control of the distance between the iterates and the identity map. Under such a $(C,\delta)$-deviation condition, we show that H\"older continuous super-Liouville irrational pseudo-rotations are $C^0$-rigid with an exponential decay rate and that $C^k$ semi-irrational pseudo-rotations of strong non-Brjuno type exhibit $C^{k-1}$-rigidity with a superpolynomial decay rate. Moreover, under this deviation condition and sufficiently large irrationality measure, we show that H\"older continuous skew products on $\mathbb{T}^2$ over circle rotations are $C^0$-rigid with a polynomial decay rate. As a consequence, all these classes satisfy Sarnak's conjecture.

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Yinshan Chang, Jian Wang, Junchang Zhou. 2026-08-27. Rigidity on the two-torus and Sarnak's conjecture. https://arxiv.org/abs/2608.26976

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