Journal of the Mathematical Society of Japan

Dominated splitting of differentiable dynamics with $\mathrm{C}^1$-topological weak-star property

Xiongping DAI

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We study weak hyperbolicity of a differentiable dynamical system which is robustly free of non-hyperbolic periodic orbits of Markus type. Let S be a $\mathrm{C}^1$-class vector field on a closed manifold $M^n$, which is free of any singularities. It is of $\mathrm{C}^1$-weak-star in case there exists a $\mathrm{C}^1$-neighborhood $\mathscr{U}$ of S such that for any X$\in\mathscr{U}$, if $P$ is a common periodic orbit of X and S with S$_{\upharpoonright P}=$X$_{\upharpoonright P}$, then $P$ is hyperbolic with respect to X. We show, in the framework of Liao theory, that S possesses the $\mathrm{C}^1$-weak-star property if and only if it has a natural and nonuniformly hyperbolic dominated splitting on the set of periodic points $\mathrm{Per}$(S), for the case $n=3$.

Article information

J. Math. Soc. Japan, Volume 64, Number 4 (2012), 1249-1295.

First available in Project Euclid: 29 October 2012

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Primary: 37C10: Vector fields, flows, ordinary differential equations
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D05: Hyperbolic orbits and sets 37C27: Periodic orbits of vector fields and flows 34D30: Structural stability and analogous concepts [See also 37C20]

dominated splitting weak-star property Liao theory


DAI, Xiongping. Dominated splitting of differentiable dynamics with $\mathrm{C}^1$-topological weak-star property. J. Math. Soc. Japan 64 (2012), no. 4, 1249--1295. doi:10.2969/jmsj/06441249.

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