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J. G. Sinaï and D. V. Anosovhave proved that the geodesic flow in the tangent bundle induces a Kolmogorov system; in particular, the flow is strongly mixing and has countable Lebesgue spectrum [ l ] , [8]. Indeed, their methods apply to many dynamical systems other than geodesic flows, and Anosov's notion of E7-system seems to give remarkably complete results in the instability theory of flows and cascades on differentiable manifolds. The extension of a weaker result (weak mixing) from the tangent to the principal bundle would hardly be worthwhile if it were not for the fact that our method seems to provide a new and powerful way of attacking global problems on manifolds with complete connections. Namely, noticing that a welldefined infinite dimensional Lie group (called the Pythagorean group, below) acts transitively on the principal bundle, we try to apply group representation ideas to its analysis. As in Anosov's proof, one can trace the evolution of our approach back to the Hedlund-Hopf technique of asymptotic geodesies. However, once it is realized that Hopf's introduction of horocyclic coordinates is, in the constant curvature case, a reflection of the Iwasawa decomposition of the associated group, a great many points in his fundamental paper [4] take on new significance. In the Lie group case, this approach has been exploited by L. Auslander, Hahn, and the present author [2], [3], and C. C. Moore [7]. (See also the earlier work by Gelfand and Fomin, and Mautner; references may be found in the above cited papers and book.) Because we use horocycle flows extensively, a result inaccessible to entropy theory does fall out.

The Pythagorean group and ergodic flows