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October, 2012 Dominated splitting of differentiable dynamics with $\mathrm{C}^1$-topological weak-star property
Xiongping DAI
J. Math. Soc. Japan 64(4): 1249-1295 (October, 2012). DOI: 10.2969/jmsj/06441249


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 $n=3$.


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Xiongping DAI. "Dominated splitting of differentiable dynamics with $\mathrm{C}^1$-topological weak-star property." J. Math. Soc. Japan 64 (4) 1249 - 1295, October, 2012.


Published: October, 2012
First available in Project Euclid: 29 October 2012

zbMATH: 1281.37009
MathSciNet: MR2998923
Digital Object Identifier: 10.2969/jmsj/06441249

Primary: 37C10
Secondary: 34D30 , 37C27 , 37D05 , 37D30

Keywords: dominated splitting , Liao theory , weak-star property

Rights: Copyright © 2012 Mathematical Society of Japan


Vol.64 • No. 4 • October, 2012
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