Generic properties of geodesic flows on analytic hypersurfaces of Euclidean space

Andrew Clarke

doi:10.3934/dcds.2022127

Article Contents

2022, Volume 42, Issue 12: 5839-5868. Doi: 10.3934/dcds.2022127

This issue Previous Article Hölder regularity for mixed local and nonlocal $ p $-Laplace parabolic equations Next Article Anderson localization for multi-frequency quasi-periodic Jacobi operators

Generic properties of geodesic flows on analytic hypersurfaces of Euclidean space

Andrew Clarke^1,

1.
Universitat Politècnica de Catalunya, Spain

Received: February 2022

Early access: September 7, 2022

Published online: September 7, 2022

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No 757802)..

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Abstract

Consider the geodesic flow on a real-analytic closed hypersurface $ M $ of $ \mathbb{R}^n $, 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:

● If $ M $ is a real-analytic closed hypersurface in $ \mathbb{R}^n $ (with $ n \geq 3 $) on which the geodesic flow with respect to the induced metric has a nonhyperbolic periodic orbit, then $ C^{\omega} $-generically the geodesic flow on $ M $ with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit; and

● There is a $ C^{\omega} $-open and dense set of real-analytic, closed, and strictly convex surfaces $ M $ in $ \mathbb{R}^3 $ on which the geodesic flow with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit.

These are among the first perturbation-theoretic results for real-analytic geodesic flows.

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Mathematics Subject Classification: Primary: 37D40, 37J46; Secondary: 37C29.

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