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

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 \m...

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Bibliographic Details
Published in:Journal of the Mathematical Society of Japan 2012, Vol.64 (4), p.1249-1295
Main Author: DAI, Xiongping
Format: Article
Language:English
Subjects:
34D30
37C10
37C27
37D05
37D30
dominated splitting
Liao theory
weak-star property
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Summary: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.
ISSN:0025-5645
1881-2333
DOI:10.2969/jmsj/06441249