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Chiyi Luo

Publications and source records attributed to Chiyi Luo.

11 recordsLinked to original sources
arXiv2026

Symbolic dynamics for non-uniformly hyperbolic flows

We construct symbolic dynamics for non-uniformly hyperbolic flows, in any dimension, possibly with fixed points. More precisely, for each $\chi>0$, we code a set which has full measure for every $\chi$-hyperbolic invariant probability measure that gives zero mass to the set of fixed points. As a main application, we prove that a three dimensional $C^\infty$ flow with positive topological entropy on a closed manifold has finitely many ergodic measures of maximal entropy. For flows in any dimension, we also provide applications to the number of periodic orbits and to the Bernoulli property for equilibrium states of H\"older continuous potentials. The main technical result of this paper is a new method for handling singularities of vector fields by modifying the Riemannian metric. This technique is analogous to a blowup, which allows many results for nonsingular vector fields to be directly applied to vector fields with singularities.

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arXiv2026

Symbolic extension of $\mathcal{C}^{1,\alpha}$ maps in arbitrary dimensions

We prove that every $\mathcal{C}^{1,\alpha}$ self-map of a compact Riemannian manifold, in arbitrary dimension, admits a symbolic extension. This gives a positive answer to the conjecture of Downarowicz and Newhouse in arbitrary dimensions. Our method is based on analyzing the small singular values of the linearized orbit operator.

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arXiv2025

Effective SPR property for surface diffeomorphisms and three-dimensional vector fields

In this paper, we prove that ergodic measures with large entropy give uniformly large measure to the set of points with simultaneously long unstable and long stable manifolds. As a consequence, for $C^{\infty}$ surface diffeomorphisms, we establish an effective version of the SPR property. For $C^{\infty}$ three-dimensional flows without singularities, we prove the finiteness of equilibrium measures for admissible potentials whose variation is strictly less than half of the topological entropy.

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arXiv2025

Upper semi-continuity of metric entropy for $\mathcal{C}^{1,\alpha}$ diffeomorphisms

We establish a uniform approximation of metric entropy by partition entropy using uniform partitions for $\cC^{1,\alpha}$ three-dimensional diffeomorphisms. This gives several consequences for diffeomorphisms on a compact manifold $M$ with ${\rm dim} M\leq 3$. First, if an invariant measure $\mu$ is a continuity point of the sum of its positive Lyapunov exponents, then $\mu$ is an upper semi-continuity point of the entropy map. Second, it provides a slight improvement of the necessary condition for strong positive recurrence of surface and three-dimensional diffeomorphisms. Third, it yields continuity of dimensions for measures of maximal entropy.

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arXiv2024

Continuity properties of ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms

Let $f$ be a $C^r$ surface diffeomorphism with large entropy (more precisely, $h_{\rm top}(f)>\lambda_{\min}(f)/{r}$). Then the number of ergodic measures of maximal entropy is upper semicontinuous at $f$. This generalizes the $C^\infty$ case studied in \cite{BCS22}, answering Question 1.9 there. Moreover, the number of such measures is locally constant if and only if every ergodic measure of maximal entropy of $f$ admits an ergodic continuation under small perturbations. In this case, the accumulation points of ergodic measures of maximal entropy are themselves ergodic. These facts are new, even in the $C^\infty$ case.

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arXiv2024

Upper semi-continuity of metric entropy for diffeomorphisms with dominated splitting

For a $C^{r}$ $(r>1)$ diffeomorphism on a compact manifold that admits a dominated splitting, this paper establishes the upper semi-continuity of the entropy map. More precisely, this paper establishes the upper semi-continuity of the entropy map in the following two cases: (1) if a sequence of invariant measures has only positive Lyapunov exponents along a sub-bundle and non-positive Lyapunov exponents along another sub-bundle, then the upper limit of their metric entropies is less than or equal to the entropy of the limiting measure; (2) if an invariant measure has positive Lyapunov exponents along a sub-bundle and non-positive Lyapunov exponents along another sub-bundle, then the entropy map is upper semi-continuous at this measure.

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arXiv2024

Ergodic measures with large entropy have long unstable manifolds for $C^\infty$ surface diffeomorphisms

We prove that for ergodic measures with large entropy have long unstable manifolds for $C^\infty$ surface diffeomorphisms. Specifically, for any $\alpha>0$, there exist constants $\beta>0$ and $c>0$ such that for every ergodic measure $\mu$ with metric entropy large than $\alpha$, the set of points with the size of unstable manifolds large than $\beta$ has $\mu$-measure large than $c$.

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arXiv2022

Characterization of SRB Measures for Random Dynamical Systems in a Banach space

This paper considers $C^2$ random dynamical systems in a Banach space, and proves that under some mild conditions, SRB measures are characterized by invariant measures satisfying Pesin's entropy formula, in which entropy is equal to the sum of positive Lyapunov exponents of the system. This can be regarded as a random version of the main result in Blumenthal and Young's paper \cite{Young17}.

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arXiv2022

Hölder continuity of Oseledets subspaces for linear cocycles on Banach spaces

Let $f:X\to X$ be an invertible Lipschitz transformation on a compact metric space $X$. Given a Hölder continuous invertible operator cocycles on a Banach space and an $f$-invariant ergodic measure, this paper establishes the Hölder continuity of Oseledets subspaces over a compact set of arbitrarily large measure. This extends a result in \cite{Simion16} for invertible operator cocycles on a Banach space. Finally, this paper proves the Hölder continuity in the non-invertible case.

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