• Communications of the Korean Mathematical Society
• /
• v.12 no.3
• /
• pp.779-788
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.36 no.1
• /
• pp.147-159
• /
• 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.