arXiv subjects

Junchang Zhou

Publications and source records attributed to Junchang Zhou.

3 recordsLinked to original sources

arXiv2026

Rigidity on the two-torus and Sarnak's conjecture

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.

math.DS↗

arXiv2026

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

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 $\tau=1$.

math.DS↗

arXiv2026

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

This note supplies genuinely non-fibred examples for the manuscript: Rigidity on the Two-Torus and Sarnak's Conjecture. For every $0<\delta<\tfrac12$, 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.

math.DS↗