arXiv subjects

Rusong Zheng

Publications and source records attributed to Rusong Zheng.

5 recordsLinked to original sources

arXiv2026

Komuro Expansivity and Periodic Orbit Growth for Multi-Singular Hyperbolic Flows

In this paper we prove that every multi-singular hyperbolic set of a $C^1$ flow is Komuro expansive. This is the strongest natural form of expansivity for flows with singularities accumulated by regular orbits, allowing arbitrary orientation-preserving time reparametrizations. We then use this to establish a two-sided asymptotic counting bound of the form $e^{ht}/t$ for periodic orbits, and prove that the normalized orbit measures converge to the unique measure of maximal entropy. We also establish the same results for a $C^1$ open and dense subset of star vector fields.

math.DS↗

arXiv2024

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 $X$.

math.DS↗

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 $Γ$ of $X_0$, we consider the closed subset $\mathscr{X}^r(M,Γ)$ of $\mathscr{X}^r(M)$, consisting of all $C^r$ vector fields which coincide with $X_0$ on $Γ$. Study of such a set naturally arises when one needs to perturb a system while keeping part of the dynamics untouched. A vector field $X\in\mathscr{X}^r(M,Γ)$ is called $Γ$-avoiding Kupka-Smale, if the dynamics away from $Γ$ is Kupka-Smale. We show that a generic vector field in $\mathscr{X}^r(M,Γ)$ is $Γ$-avoiding Kupka-Smale. In the $C^1$ topology, we obtain more generic properties for $\mathscr{X}^1(M,Γ)$. With these results, we further study codimension one partially hyperbolic dynamics for generic vector fields in $\mathscr{X}^1(M,Γ)$, 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.

math.DS↗

arXiv2023

An example derived from Lorenz attractor

We consider a DA-type surgery of the famous Lorenz attractor in dimension 4. This kind of surgeries have been firstly used by Smale [S] and Mañé [M1] to give important examples in the study of partially hyperbolic systems. Our construction gives the first example of a singular chain recurrence class which is Lyapunov stable, away from homoclinic tangencies and exhibits robustly heterodimensional cycles. Moreover, the chain recurrence class has the following interesting property: there exists robustly a 2-dimensional sectionally expanding subbundle (containing the flow direction) of the tangent bundle such that it is properly included in a subbundle of the finest dominated splitting for the tangent flow.

math.DS↗

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.

math.DS↗