Generic properties of geodesic flows on analytic hypersurfaces of Euclidean space

Abstract

<p style='text-indent:20px;'>Consider the geodesic flow on a real-analytic closed hypersurface <inline-formula><tex-math id="M1">\begin{document}$ M $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, equipped with the induced metric. How commonly can we expect such flows to have a transverse homoclinic orbit? In this paper, we give the following two partial answers to this question:</p><p style='text-indent:20px;'>● If <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> is a real-analytic closed hypersurface in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> (with <inline-formula><tex-math id="M5">\begin{document}$ n \geq 3 $\end{document}</tex-math></inline-formula>) on which the geodesic flow with respect to the induced metric has a nonhyperbolic periodic orbit, then <inline-formula><tex-math id="M6">\begin{document}$ C^{\omega} $\end{document}</tex-math></inline-formula>-generically the geodesic flow on <inline-formula><tex-math id="M7">\begin{document}$ M $\end{document}</tex-math></inline-formula> with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit; and</p><p style='text-indent:20px;'>● There is a <inline-formula><tex-math id="M8">\begin{document}$ C^{\omega} $\end{document}</tex-math></inline-formula>-open and dense set of real-analytic, closed, and strictly convex surfaces <inline-formula><tex-math id="M9">\begin{document}$ M $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M10">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula> on which the geodesic flow with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit.</p><p style='text-indent:20px;'>These are among the first perturbation-theoretic results for real-analytic geodesic flows.</p>