Trajectories on Nonorientable Two-Dimensional Manifolds

Abstract

A model of two-dimensional grain growth based on growth and shrinkage of two sub-populations of grains is presented. The model allows a natural description of self-similar normal grain growth. Results from numerical solution of the time-dependent model and the time-independent self-similar reduction are given.

On symmetric logarithm and some old examples in smooth ergodic theory

“…For the first two surfaces the absence of ergodic smooth flows follows from the Poincaré-Bendixson theorem. The absence of ergodic smooth flows on the Klein bottle was proved by Aranson [3] and Markley [19]. Moreover, as shown by Kochergin in [14], on any surface different from the sphere, the projective plane, and the Klein bottle, there is a mixing C ∞ -flow preserving a positive C ∞ -measure.…”

Empty Interior Recurrence for Continuous Flows on Surfaces

“…Together with the sphere S 2 , the projective plane P 2 and the Klein B 2 are the easiest surfaces to work with because they admit no flows having nontrivial recurrent orbits [Aranson, 1969], [Thomas, 1970], [Markley, 1969], [Gutierrez, 1978]. In we extended the previous results by characterizing ω-limit sets of nonrecurrent orbits in arbitrary surfaces.…”

A topological characterization of the ω-limit sets for analytic flows on the plane, the sphere and the projective plane

“…In the case S ⊂ S 2 the classical Poincaré-Bendixson argument (every ω-limit set must intersect every transversal to the flow at most at one point) still works because of the flow box theorem. In the projective plane case, independent proofs were first provided by Aranson in 1969 andThomas in 1970 [3,50]. The stronger result below may have escaped unnoticed.…”

“…We recall that, together with the sphere, the only compact surfaces for which there are no flows with nontrivial recurrences are the projective plane and the Klein Bottle [3,18,33,50].…”