arXiv subjects

Ruihao Gu

Publications and source records attributed to Ruihao Gu.

9 recordsLinked to original sources

arXiv2026

On the Absence of Anosov Factors for DA Local Diffeomorphisms

We give a class of local diffeomorphisms which are homotopic to toral hyperbolic endomorphisms, but which are not topologically semi-conjugate, within the homotopic class of the identity, to any Anosov local diffeomorphism. In particular, we show that for a non-invertible partially hyperbolic $C^1$-smooth local diffeomorphism with expanding directions on the $2$-torus, if it is semi-conjugate to an Anosov local diffeomorphism, then it is also an Anosov local diffeomorphism.

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arXiv2026

The Teichm\"uller Space of a 3-Dimensional Anosov Flow

For a transitive Anosov flow $\Phi$ on 3-dimensional closed manifold $M$ , we realize its Teichm\"uller space in the sense of smooth orbit-equivalence classes as a product of two function spaces. As an application, we show the path-connectedness of the orbit-equivalence space of 3-dimensional transitive Anosov flows which gives a positive answer of Potrie [53, Question 1] in dimension 3. Further, in the space of $C^r$-smooth ($r\geq 1$) 3-dimensional Anosov flows on $M$, we show that $\mathcal{A}^r(\Phi)$ the path component containing $\Phi$ is homotopy equivalent to the identity component of the diffeomorphism group of the manifold, namely, \[ \mathcal{A}^r(\Phi)\simeq {\rm Diff}^r_0(M). \] Moreover, we show the rigidity of time-preserving conjugacy for 3-dimensional transitive Anosov flows admitting $C^1$-smooth strong stable foliations, which gives partial answer of Gogolev-Leguil- Rodriguez Hertz [27, Question 2.8].

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arXiv2024

Stable Lyapunov Spectrum Rigidity of Nilmanifold Endomorphisms

Under some non-invertibility and irreducibility condition, for nilmanifold Anosov maps with one-dimensional stable bundle, we get the equivalence among the existence of invariant unstable bundle, the existence of topological conjugacy to its linear part, and a constant periodic stable Lyapunov exponent.

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arXiv2023

Semi-conjugacy rigidity for endomorphisms derived from Anosov on the 2-torus

Let $f$ be a non-invertible partially hyperbolic endomorphism on $\TT^2$ which is derived from a non-expanding Anosov endomorphism. Differing from the case of diffeomorphisms derived from Anosov automorphisms, there is no a priori semi-conjugacy between $f$ and its linearization on $\TT^2$. We show that $f$ is semi-conjugate to its linearization if and only if $f$ admits a partially hyperbolic splitting with two $Df$-invariant subbundles. Moreover, if we assume that $f$ has an unstable subbundle, then the semi-conjugacy is exactly a topological conjugacy, and the center Lyapunov exponents of the periodic points of $f$ coincide with its linearization. In particular, $f$ is an Anosov endomorphism and the conjugacy is smooth along the stable foliation. For the case that $f$ has a stable subbundle, there is still some rigidity in its stable Lyapunov exponents. However, we also give examples which admit a partially hyperbolic splitting with center subbundle but the semi-conjugacy is indeed non-injective. Finally, we present some applications under the volume-preserving assumption.

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arXiv2023

Smooth Stable Foliations of Anosov Diffeomorphisms

In this paper, we focus on the rigidity of $C^{2+}$-smooth codimension-one stable foliations of Anosov diffeomorphisms. Specifically, we show that if the regularity of these foliations is slightly bigger than $2$, then they will have the same smoothness as the diffeomorphisms.

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arXiv2023

Stable ergodicity of skew product endomorphisms on $T^2$

For a family of skew product endomorphisms on closed surface $f_{\phi}(x,y)=\big(l x\ , \ y+\phi(x)\big)$, where $l\in \mathbb{N}_{\geq 2}$ and $\phi: S^1\to \mathbb{R}$ is a $C^r\ (r>1)$ function, we get a dichotomy on the cohomology class of $\phi$ and the $C^1$-stable ergodicity of $f_{\phi}$, where the perturbation may not be a skew product endomorphism.

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arXiv2022

Topological and smooth classification of Anosov maps on torus

In this paper, we give a complete topological and smooth classification of non-invertible Anosov maps on torus. We show that two non-invertible Anosov maps on torus are topologically conjugate if and only if their corresponding periodic points have the same Lyapunov exponents on the stable bundles. As a corollary, if two $C^r$ non-invertible Anosov maps on torus are topologically conjugate, then the conjugacy is $C^r$-smooth along the stable foliation. Moreover, we show that the smooth conjugacy class of a non-invertible Anosov map on torus is completely determined by the Jacobians of return maps at periodic points.

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arXiv2022

Rigidity of center Lyapunov exponents for Anosov diffeomorphisms on 3-torus

Let f and g be two Anosov diffeomorphisms on T3 with three-subbundles partially hyperbolic splittings where the weak stable subbundles are considered as center subbundles. Assume that f is conjugate to g and the conjugacy preserves the strong stable foliation, then their center Lyapunov exponents of corresponding periodic points coincide. This is the converse of the main result of Gogolev and Guysinsky in [9]. Moreover, we get the same result for partially hyperbolic diffeomorphisms derived from Anosov on T3.

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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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