A surface with positive curvature and positive topological entropy
It is known, by the work of Donnay and Burns-Gerber, that there exist smooth metrics on the 2-sphere whose associated geodesic flow is Bernoulli. In the known examples, this behaviour is due to the existence of negative curvature on part of the surface. It is an open question whether there exist examples of a 2-sphere with a positively curved metric, for which the geodesic flow has complicated dynamical behaviour, for example ergodic, positive Liouville entropy, etc. In this paper, the authors give a positive answer to a question of this kind, namely, they construct real analytic Riemannian metrics on the 2-sphere with positive Gaussian curvature whose geodesic flows have positive topological entropy. Smooth metrics are constructed as small perturbations of the ellipsoid. The methods are extensions of methods due to Poincaré. The results follow from the existence of horseshoes in the dynamics of the geodesic flow. Real analytic examples follow by using approximation results.
scientific article; zbMATH DE number 590829