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For Morse-Smale diffeomorphisms with three fixed points, one proves that the closure of separatrices are flat spheres provided the dimension of manifold is not less than six, and can be wildly embedded spheres provided the dimension of manifold is four.
We obtain a nontrivial estimate of the variance of the sum of bounded partial quotients appearing in the continued-fraction expansion of a rational number with fixed denominator. As a consequence, we obtain a law of large numbers for the sum of all partial quotients.
It is proved that any SOо(1, d)-valued cocycle over an ergodic (probability) measurepreserving automorphism is cohomologous to a cocycle having one of three special forms; the recurrence property of such cocycles is also studied.