Weiss, Howard Non-smooth geodesic flows and the earthquake flow on Teichmüller space. (English) Zbl 0692.57004 Ergodic Theory Dyn. Syst. 9, No. 3, 571-586 (1989). The author shows that earthquake semi-flows on any Teichmüller space \({\mathcal T}\) are not \({\mathcal C}^ 3\) smooth, where the flows are parametrized as in a paper of S. Kerckhoff [Ann. Math., II. Ser. 117, 235-265 (1983; Zbl 0528.57008)]. This is accomplished by deriving the differential equations governing the associated semi-flows on the tangent bundle T\({\mathcal T}\) to \({\mathcal T}\) in terms of quantities \(\Gamma^ k_{ij}\) which transform as Christoffel symbols but are defined on T\({\mathcal T}\) itself. An elementary argument (abstracted to the setting of certain flows with potentially low regularity) shows that \({\mathcal C}^ 2\) smoothness of the infinitesimal generator implies that the \(\Gamma^ k_{ij}\) depend only on the underlying point of \({\mathcal T}\). The assumption of \({\mathcal C}^ 2\) smoothness then allows the explicit computation of the \(\Gamma^ k_{ij}\) for the cases of the once-punctured torus and the torus-minus-a-disk; certain earthquake flows for these surfaces are found to violate convexity of hyperbolic lengths under earthquakes [see S. Kerckhoff, loc. cit.], and the proof for a general surface follows directly from the latter case. Certain standard material on measured geodesic laminations and earthquakes is briefly surveyed. Reviewer: R.Penner Cited in 1 Document MSC: 57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53D25 Geodesic flows in symplectic geometry and contact geometry 57R30 Foliations in differential topology; geometric theory 57R50 Differential topological aspects of diffeomorphisms 30F99 Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Keywords:earthquake semi-flows on Teichmüller space; measured geodesic laminations Citations:Zbl 0528.57008 PDFBibTeX XMLCite \textit{H. Weiss}, Ergodic Theory Dyn. Syst. 9, No. 3, 571--586 (1989; Zbl 0692.57004)