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Zeya Mi

Publications and source records attributed to Zeya Mi.

12 recordsLinked to original sources
arXiv2025

Stable manifolds, Horseshoes and Lyapunov exponents for $C^1$ diffeomorphisms without domination

We develop the nonuniformly hyperbolic theory for $C^1$ diffeomorphisms admitting continuous invariant splitting without domination. This framework includes stable manifold theorems, shadowing and closing lemmas, the existence of horseshoes and the approximation of Lyapunov exponents. The foundation is a new family of resonance blocks, each arising as the forward limit set of a typical point at carefully chosen resonance times where expansion, contraction and a weak scale-dependent domination coexist.

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arXiv2025

Variety of physical measures in partially hyperbolic systems with multi 1-D centers

We provide two robust examples of globally partially hyperbolic systems with a multi one-dimensional center splitting, for which all Gibbs u-states are hyperbolic and the number of physical measures is fixed. In the second example, the physical measures have different unstable indices, and the supports of those with index at least 2 can be adjusted to lie on a non-uniformly hyperbolic set using the techniques developed for the first example. Such partially hyperbolic systems were previously studied in \cite{MC25, Bur25}, where the existence and finiteness of physical measures were established under the assumption of hyperbolicity of the Gibbs $u$-states or the SRB measures, respectively.

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arXiv2023

Physical measures on partially hyperbolic diffeomorphisms with multi 1-D centers

In this paper, we study physical measures for partially hyperbolic diffeomorphisms with multi one-dimensional centers under the condition that all Gibbs $u$-states are hyperbolic. We prove the finiteness of ergodic physical measures. Then by building a criterion for the basin covering property of physical measures, we obtain the basin covering property for ergodic physical measures when there exists some limit measure of empirical measures for Lebesgue almost every point that admits the same sign of Lyapunov exponents on each center.

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arXiv2023

Physical measures for partially hyperbolic diffeomorphisms with mixed hyperbolicity

We study the partially hyperbolic diffeomorphims whose center direction admits the u-definite property in the sense that all the central Lyapunov exponents of each ergodic Gibbs u-state are either all positive or all negative. We prove that for this kind of partially hyperbolic diffeomorphisms, there are finitely many physical measures, whose basins cover a full Lebesgue measure subset of the ambient space.

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arXiv2023

Finiteness of physical measures for diffeomorphisms with multi 1-D centers

Let $f$ be a $C^2$ diffeomorphism on compact Riemannian manifold $M$ with partially hyperbolic splitting $$ TM=E^u\oplus E_1^c\oplus\cdots\oplus E_k^c \oplus E^s, $$ where $E^u$ is uniformly expanding, $E^s$ is uniformly contracting, and ${\rm dim}E_i^c=1,~ 1\le i \le k, ~k\ge 1$. We prove the finiteness of ergodic physical(SRB) measures of $f$ under the hyperbolicity of Gibbs $u$-states.

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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.

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arXiv2020

Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers

For partially hyperbolic diffeomorphisms with mostly expanding and mostly contracting centers, we establish a topological structure, called skeleton{a set consisting of finitely many hyperbolic periodic points with maximal cardinality for which there exist no heteroclinic intersections. We build the one-to-one corresponding between periodic points in any skeleton and physical measures. By making perturbations on skeletons, we study the continuity of physical measures with respect to dynamics under $C^1$-topology.

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arXiv2018

SRB measures for some diffeomorphisms with dominated splittings as zero noise limits

In this paper, we provide a technique result on the existence of Gibbs cu-states for diffeomorphisms with dominated splittings. More precisely, for given $C^2$ diffeomorphim $f$ with dominated splitting $T_ΛM = E\oplus F$ on an attractor $Λ$, by considering some suitable random perturbation of $f$, we show that for any zero noise limit of ergodic stationary measures, if it has positive integrable Lyapunov exponents along invariant sub-bundle $E$, then its ergodic components contain Gibbs cu-states associated to $E$. With this technique, we show the existence of SRB measures and physical measures for some systems exhibitting dominated splittings and weak hyperbolicity.

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arXiv2018

On the abundance of SRB measures

We prove the abundance of Sinai-Ruelle-Bowen measures for diffeomorphisms away from ones with a homoclinic tangency. This is motivated by conjectures of Palis on the existence of physical (Sinai-Ruelle-Bowen) measures for global dynamics. The main novelty in this paper is that we have to deeply study Gibbs $cu$-states in different levels. Note that we have to use random perturbations to give some upper bound of the level of Gibbs $cu$-states.

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arXiv2015

SRB measures for diffeomorphisms with continuous invariant splittings

We study the existence of SRB measures of C 2 diffeomorphisms for attractors whose bundles admit Holder continuous invariant (non-dominated) splittings. We prove the existence when one subbundle has the non-uniform expanding property on a set with positive Lebesgue measure and the other subbundle admits non-positive Lyapunov exponents on a total probability set.

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