• Title, Summary, Keyword: uniform hyperbolicity

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ROBUSTLY SHADOWABLE CHAIN COMPONENTS OF C1 VECTOR FIELDS

  • Lee, Keonhee;Le, Huy Tien;Wen, Xiao
    • Journal of the Korean Mathematical Society
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    • v.51 no.1
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    • pp.17-53
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    • 2014
  • Let ${\gamma}$ be a hyperbolic closed orbit of a $C^1$ vector field X on a compact boundaryless Riemannian manifold M, and let $C_X({\gamma})$ be the chain component of X which contains ${\gamma}$. We say that $C_X({\gamma})$ is $C^1$ robustly shadowable if there is a $C^1$ neighborhood $\mathcal{U}$ of X such that for any $Y{\in}\mathcal{U}$, $C_Y({\gamma}_Y)$ is shadowable for $Y_t$, where ${\gamma}_Y$ denotes the continuation of ${\gamma}$ with respect to Y. In this paper, we prove that any $C^1$ robustly shadowable chain component $C_X({\gamma})$ does not contain a hyperbolic singularity, and it is hyperbolic if $C_X({\gamma})$ has no non-hyperbolic singularity.

THE MEASURE OF THE UNIFORMLY HYPERBOLIC INVARIANT SET OF EXACT SEPARATRIX MAP

  • Kim, Gwang-Il;Chi, Dong-Pyo
    • Communications of the Korean Mathematical Society
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    • v.12 no.3
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    • pp.779-788
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    • 1997
  • In this work, using the exact separatrix map which provides an efficient way to describe dynamics near the separatrix, we study the stochastic layer near the separatrix of a one-degree-of-freedom Hamilitonian system with time periodic perturbation. Applying the twist map theory to the exact separatrix map, T. Ahn, G. I. Kim and S. Kim proved the existence of the uniformly hyperbolic invariant set(UHIS) near separatrix. Using the theorems of Bowen and Franks, we prove this UHIS has measure zero.

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ELLIPTIC BIRKHOFF'S BILLIARDS WITH $C^2$-GENERIC GLOBAL PERTURBATIONS

  • Kim, Gwang-Il
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.147-159
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    • 1999
  • Tabanov investigated the global symmetric perturbation of the integrable billiard mapping in the ellipse [3]. He showed the nonintegrability of the Birkhoff billiard in the perturbed domain by proving that the principal separatrices splitting angle is not zero.In this paper, using the exact separatrix map of an one-degree-of freedom Hamiltoniam system with time periodic perturbation, we show the existence the stochastic layer including the uniformly hyperbolic invariant set which implies the nonintegrability near the separatrices of a Birkhoff's billiard in the domain bounded by $C^2$ convex simple curve constructed by the generic global perturbation of the ellipse.

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