arXiv subjects

Yuntao Zang

Publications and source records attributed to Yuntao Zang.

6 recordsLinked to original sources

arXiv2026

Entropy formula for surface diffeomorphisms

Let $f$ be a $C^r$ ($r>1$) diffeomorphism on a compact surface $M$ with $h_{\rm top}(f)\geq\frac{λ^{+}(f)}{r}$ where $λ^{+}(f):=\lim_{n\to+\infty}\frac{1}{n}\max_{x\in M}\log \left\|Df^{n}_{x}\right\|$. We establish an equivalent formula for the topological entropy: $$h_{\rm top}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int_{M}\left\|Df^{n}_{x}\right\|\,dx.$$ We also characterize the topological entropy via the volume growth of curves and several applications are presented. Our approach builds on the key ideas developed in the works of Buzzi-Crovisier-Sarig (\emph{Invent. Math.}, 2022) and Burguet (\emph{Ann. Henri Poincaré}, 2024) concerning the continuity of the Lyapunov exponents.

math.DS↗

arXiv2025

Measures of maximal entropy for $C^\infty$ three-dimensional flows

We prove for $C^\infty$ non-singular flows on three-dimensional compact manifolds with positive entropy, there are at most finitely many ergodic measures of maximal entropy. This result extends the notable work of Buzzi-Crovisier-Sarig (\emph{Ann. of Math.}, 2022) on surface diffeomorphisms. Our approach differs by addressing the continuity of Lyapunov exponents and the uniform largeness of Pesin sets for measures of maximal entropy. Furthermore, it also provides an alternative proof for the case of surface diffeomorphisms.

math.DS↗

arXiv2020

Volume growth and topological entropy of certain partially hyperbolic systems

Let $f$ be a $C^{1}$ diffeomorphism on a compact manifold $M$ admitting a partially hyperbolic splitting $TM=E^{s}\oplus_{\prec} E^{1}\oplus_{\prec} E^{2}\cdots \oplus_{\prec}E^{l}\oplus_{\prec} E^{u}$ where $E^{s}$ is uniformly contracting, $E^{u}$ is uniformly expanding and $\dim E^{i}=1,\,1\leq i\leq l.$ We prove an entropy formula w.r.t. the volume growth rate of subspaces in the tangent bundle: $$h_{\rm{top}}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int\max_{V\subset T_{x}M}|\det Df_{x}^{n}|_{V}|\,d x.$$

math.DS↗

arXiv2020

SRB measures for partially hyperbolic flows with mostly expanding center

We prove that a partially hyperbolic attractor for a $C^1$ vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is $C^2$, and the center bundle admits the sectional expanding condition w.r.t. any Gibbs $u$-state, then the attractor can only support finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin. The proof of these results has to deal with the difficulties which do not occur in the case of diffeomorphisms.

math.DS↗

arXiv2020

Entropies and volume growth of unstable manifolds

Let $f$ be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $μ$. We relate those entropies to covering numbers in order to give a new upper bound on the metric entropy of $μ$ in terms of Lyapunov exponents and topological entropy or volume growth of sub-manifolds. We also discuss extensions to the $C^{1+α},\,α>0$ case.

math.DS↗

arXiv2019

On the volume of sectional-hyperbolic sets

For a transitive sectional-hypebolic set $Λ$ with positive volume on a $d$-dimensional manifold $M$($d\ge3$), we show that $Λ=M$ and $Λ$ is a uniformly hyperbolic set without singularities

math.DS↗