We prove that the rotation in time T of a trajectory of a K-Lipschitz vector ¯eld in Rn around a given point (stationary or non-stationary) is bounded by A + BKT with A;B absolute constants. In particular, trajectories of a Lipschitz vector ¯eld in ¯nite time cannot have an in¯nite rotation around a given point (while trajectories of a C1 vector ¯eld may have an in¯- nite rotation around a straight line in ¯nite time). The bound above extends to the mutual rotation of two trajectories (for the time intervals T and T0, respectively) of a K-Lipschitz vector ¯eld in R3: this rotation is bounded from above by the quantity CK min(T; T0) + DK2TT0.

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We prove that if a curve of a non-isotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety then it is special. This result fits into the context of Zilber-Pink conjecture and partially generalizes a result of Faltings. Moreover by using the polyhedral reduction theory we give a new proof of a result of Bertrand.

We obtain improved fractional Poincaré inequalities in John domains of a metric space (X,d) endowed with a doubling measure μ under some mild regularity conditions on the measure μ. We also give sufficient conditions on a bounded domain to support fractional Poincaré type inequalities in this setting.