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Shaobo Gan

Publications and source records attributed to Shaobo Gan.

At least 19 recordsLinked to original sources
arXiv2026

SRB Measures for $C^{1+\mathrm{Dini}}$ Diffeomorphisms

For $C^{1+\mathrm{Dini}}$ diffeomorphisms, we prove a Ledrappier--Young-type characterization of SRB measures among invariant measures whose supports admit dominated splittings. When the dominating bundle has only positive Lyapunov exponents, absolute continuity of the conditional measures on the corresponding Pesin unstable manifolds is equivalent to the partial Pesin entropy formula. As an application, we obtain SRB measures for partially hyperbolic mostly expanding attractors in the $C^{1+\mathrm{Dini}}$ category. Counterexamples are provided to show that these conclusions fail in the $C^1$ category, even under uniform hyperbolicity.

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arXiv2025

Ledrappier-Young entropy formula for $C^1$ diffeomorphisms with dominated splitting Part 1: Unstable entropy formula and invariance principle

We study the unstable entropy of $C^1$ diffeomorphisms with dominated splittings. Our main result shows that when the zero Lyapunov exponent has multiplicity one, the center direction contributes no entropy, and the unstable entropy coincides with the metric entropy. This extends the celebrated work of Ledrappier-Young [18] for $C^2$ diffeomorphisms to the $C^1$ setting under these assumptions. In particular, our results apply to $C^1$ diffeomorphisms away from homoclinic tangencies due to [20]. As consequences, we obtain several applications at $C^1$ regularity. The Avila-Viana invariance principle [7, 33] holds when the center is one-dimensional. Results on measures of maximal entropy due to Hertz-Hertz-Tahzibi-Ures [25], Tahzibi-Yang [33], and Ures-Viana-Yang-Yang [34, 35] also remain valid for $C^1$ diffeomorphisms.

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arXiv2024

Generic properties of vector fields identical on a compact set and codimension one partially hyperbolic dynamics

Let $\mathscr{X}^r(M)$ be the set of $C^r$ vector fields on a boundaryless compact Riemannian manifold $M$. Given a vector field $X_0\in\mathscr{X}^r(M)$ and a compact invariant set $\Gamma$ of $X_0$, we consider the closed subset $\mathscr{X}^r(M,\Gamma)$ of $\mathscr{X}^r(M)$, consisting of all $C^r$ vector fields which coincide with $X_0$ on $\Gamma$. Study of such a set naturally arises when one needs to perturb a system while keeping part of the dynamics untouched. A vector field $X\in\mathscr{X}^r(M,\Gamma)$ is called $\Gamma$-avoiding Kupka-Smale, if the dynamics away from $\Gamma$ is Kupka-Smale. We show that a generic vector field in $\mathscr{X}^r(M,\Gamma)$ is $\Gamma$-avoiding Kupka-Smale. In the $C^1$ topology, we obtain more generic properties for $\mathscr{X}^1(M,\Gamma)$. With these results, we further study codimension one partially hyperbolic dynamics for generic vector fields in $\mathscr{X}^1(M,\Gamma)$, giving a dichotomy of hyperbolicity and Newhouse phenomenon. As an application, we obtain that $C^1$ generically in $\mathscr{X}^1(M)$, a non-trivial Lyapunov stable chain recurrence class of a singularity which admits a codimension 2 partially hyperbolic splitting with respect to the tangent flow is a homoclinic class.

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arXiv2023

On the density of Birkhoff sums for Anosov diffeomorphisms

Let f be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides of zero, then the set of Birkhoff sums of all periodic points is dense in the whole set of real numbers.

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arXiv2022

Rigidity of stable Lyapunov exponents and integrability for Anosov maps

Let $f$ be a non-invertible irreducible Anosov map on $d$-torus. We show that if the stable bundle of $f$ is one-dimensional, then $f$ has the integrable unstable bundle, if and only if, every periodic point of $f$ admits the same Lyapunov exponent on the stable bundle with its linearization. For higher-dimensional stable bundle case, we get the same result on the assumption that $f$ is a $C^1$-perturbation of a linear Anosov map with real simple Lyapunov spectrum on the stable bundle. In both cases, this implies if $f$ is topologically conjugate to its linearization, then the conjugacy is smooth on the stable bundle.

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arXiv2022

Lyapunov stable chain recurrence classes for singular flows

We show that for a $C^1$ generic vector field $X$ away from homoclinic tangencies, a nontrivial Lyapunov stable chain recurrence class is a homoclinic class. The proof uses an argument with $C^2$ vector fields approaching $X$ in $C^1$ topology, with their Gibbs $F$-states converging to a Gibbs $F$-state of $X$.

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arXiv2021

$C^r$-Closing lemma for partially hyperbolic diffeomorphisms with 1D-center bundle

For every $r\in\mathbb{N}_{\geq2}\cup\{\infty\}$, we prove the $C^r$-closing lemma for general and conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle. In particular, it implies periodic points are dense for $C^r$-generic conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle.

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arXiv2020

Statistical properties of physical-like measures

In this paper we consider the semi-continuity of the physical-like measures for diffeomorphisms with dominated splittings. We prove that any weak-* limit of physical-like measures along a sequence of $C^1$ diffeomorphisms $\{f_n\}$ must be a Gibbs $F$-state for the limiting map $f$. As a consequence, we establish the statistical stability for the $C^1$ perturbation of the time-one map of three-dimensional Lorenz attractors, and the continuity of the physical measure for the diffeomorphisms constructed by Bonatti and Viana.

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arXiv2020

Partially volume expanding diffeomorphisms

We call a partially hyperbolic diffeomorphism \emph{partially volume expanding} if the Jacobian restricted to any hyperplane that contains the unstable bundle $E^u$ is larger than $1$. This is a $C^1$ open property. We show that any $C^{1+}$ partially volume expanding diffeomorphisms admits finitely many physical measures, the union of whose basins has full volume.

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arXiv2019

Rigidity of center Lyapunov exponents and $su$-integrability

Let $f$ be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism $A$ on $\mathbb{T}^3$. We show that the stable and unstable bundles of $f$ are jointly integrable if and only if every periodic point of $f$ admits the same center Lyapunov exponent with $A$. In particular, $f$ is Anosov. Thus every conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism on $\mathbb{T}^3$, is ergodic. This proves the Ergodic Conjecture proposed by Hertz-Hertz-Ures on $\mathbb{T}^3$.

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arXiv2017

A robustly transitive diffeomorphism of Kan's type

We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admitting two physical measures with intermingled basins. In particularly, all these diffeomorphisms are not topologically mixing. Moreover, for every such example, it exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.

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arXiv2015

Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

We prove that, for $C^1$-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-hyperbolic ergodic measure supported on it. This proves a conjecture by Díaz and Gorodetski [28]. We also discuss the conjectured existence of periodic points with different stable dimension in the class.

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arXiv2014

On the singular hyperbolicity of star flows

We prove for a generic star vector field $X$ that, if for every chain recurrent class $C$ of $X$ all singularities in $C$ have the same index, then the chain recurrent set of $X$ is singular hyperbolic. We also prove that every Lyapunov stable chain recurrent class of $X$ is singular hyperbolic. As a corollary, we prove that the chain recurrent set of a generic 4-dimensional star flow is singular hyperbolic.

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