arXiv subjects

Jiesong Zhang

Publications and source records attributed to Jiesong Zhang.

5 recordsLinked to original sources

arXiv2026

Invariant complex structures for affine automorphisms: a cocycle viewpoint

We prove that if a holomorphic diffeomorphism of a compact complex manifold is bi-Lipschitz conjugate to an ergodic affine automorphism $A$ on $Γ\backslash G$, then the conjugacy is $C^\infty$. Moreover, if $A$ is weakly mixing, then the induced complex structure on $Γ\backslash G$ is left-invariant. As applications, we establish a regularity bootstrap result for holomorphic Anosov diffeomorphisms bi-Lipschitz conjugate to affine models, as well as a holomorphic analogue of the rigidity theorem for higher-rank abelian Anosov actions by Hertz--Wang. The key observation is that the condition for a diffeomorphism to preserve a complex structure has the same form as the cocycle compatibility relation appearing in the study of centralizers. This places invariant complex structures and centralizers within a common $\mathbb Z^2$-cocycle framework. From this viewpoint, our main result may be regarded as a holomorphic counterpart of the Lipschitz centralizer rigidity theorem of Damjanović--Wilkinson--Wu--Xu for affine automorphisms.

math.DS↗

arXiv2026

Extremal distributions of partially hyperbolic systems: the Lipschitz threshold

We prove a sharp phase transition in the regularity of the extremal distribution $E^s \oplus E^u$ for $C^\infty$ volume-preserving partially hyperbolic diffeomorphisms on closed $3$-manifolds: if $E^s \oplus E^u$ is Lipschitz, then it is automatically $C^\infty$. This extends the rigidity phenomenon established by Foulon--Hasselblatt for conservative Anosov flows in dimension $3$ to the partially hyperbolic setting. This gain in regularity has several applications to rigidity problems. In particular, we study the relationship between the $\ell$-integrability condition introduced by Eskin--Potrie--Zhang and joint integrability in the conservative setting, yielding rigidity results for $u$-Gibbs measures. We also obtain several $C^\infty$ classification results for partially hyperbolic diffeomorphisms on $3$-manifolds under various assumptions.

math.DS↗

arXiv2025

On holomorphic partially hyperbolic systems

We construct examples illustrating that dynamically-defined distributions of holomorphic diffeomorphisms on compact complex manifolds are not necessarily holomorphic in any open subset. More precisely, for any $n\geq 5$, we construct a holomorphic fibered partially hyperbolic system on a complex $n$-fold, where the center distribution is not holomorphic in any open subset. For $n=3$ we demonstrate a contrast: the center distribution of any fibered holomorphic partially hyperbolic diffeomorphism on a complex $3$-fold is holomorphic. In particular, any such a system is a holomorphic skew product over a linear automorphism on a complex $2$-torus.

math.DS↗

arXiv2025

Invariant distributions of partially hyperbolic systems: fractal graphs, excessive regularity, and rigidity

We introduce a novel approach linking fractal geometry to partially hyperbolic dynamics, revealing several new phenomena related to regularity jumps and rigidity. One key result demonstrates a sharp phase transition for partially hyperbolic diffeomorphisms $f \in \mathrm{Diff}^\infty_{\mathrm{vol}}(\mathbb{T}^3)$ with a contracting center direction: $f$ is $C^\infty$-rigid if and only if both $E^s$ and $E^c$ exhibit Hölder exponents exceeding the expected threshold. Specifically, we prove: If the Hölder exponent of $E^s$ exceeds the expected value, then $E^s$ is $C^{1+}$ and $E^u \oplus E^s$ is jointly integrable. If the Hölder exponent of $E^c$ exceeds the expected value, then $W^c$ forms a $C^{1+}$ foliation. If $E^s$ (or $E^c$) does not exhibit excessive Hölder regularity, it must have a fractal graph. These and related results originate from a general non-fractal invariance principle: for a skew product $F$ over a partially hyperbolic system $f$, if $F$ expands fibers more weakly than $f$ along $W^u_f$ in the base, then for any $F$-invariant section, if $Φ$ has no a fractal graph, then it is smooth along $W^u_f$ and holonomy-invariant. Motivated by these findings, we propose a new conjecture on the stable fractal or stable smooth behavior of invariant distributions in typical partially hyperbolic diffeomorphisms.

math.DS↗

arXiv2024

On uniformly quasiconformal Anosov diffeomorphisms with two dimensional distributions

We prove that a transitive uniformly $u$-quasiconformal Anosov diffeomorphism with a two-dimensional unstable distribution has a globally defined stable holonomy. As a corollary, we are able to remove an additional assumption in a theorem of Kalinin-Sadovskaya, and deduce that all transitive uniformly quasiconformal Anosov diffeomorphisms are $C^{\infty}$-conjugate to affine Anosov diffeomorphisms on infra-torus.

math.DS↗