EMPTY INTERIOR RECURRENCE FOR CONTINUOUS FLOWS ON SURFACES

In this paper we characterize topologically the empty interior subsets of a compact surface S which can be ω-limit sets of recurrent orbits (but not of nonrecurrent ones) of continuous flows on S. This culminates the classification of ω-limit sets for surface flows initiated in [Jiménez & Soler, 2001; Soler, 2003; Jiménez & Soler, 2004a, 2004b]. We also show that this type of ω-limit sets can always be realized (up to topological equivalence) by smooth flows but cannot be realized by analytic flows.

PERIODICITY AND RECURRENCE IN PIECEWISE ROTATIONS OF EUCLIDEAN SPACES

We consider the behavior of piecewise isometries in Euclidean spaces. We show that if n is odd and the system contains no orientation reversing isometries then recurrent orbits with rational coding are not expected. More precisely, a prevalent set of piecewise isometries do not have recurrent points having rational coding. This implies that when all atoms are convex no periodic points exist for almost every piecewise isometry. By contrast, if n≥2 is even then periodic points are stable for almost every piecewise isometry whose set of defining isometries are not orientation reversing. If, in addition, the defining isometries satisfy an incommensurability condition then all unbounded orbits must be irrationally coded.

Recurrence in Lipschitz stable flows

The purpose of this paper is to get some necessary conditions for a Poisson stable flow to be recurrent and to analyse the bilateral versions of positive and negative Lipschitz stability. Moreover, a characterisation of recurrent orbits is obtained in a certain flow.

Existence of periodic orbits of autonomous ordinary differential equations

SynopsisThe Poincaré-Bendixson theorem, concerning the existence of periodic orbits of plane autonomous systems, is extended to higher order systems under certain conditions. Under similar conditions, a complementary theorem on the existence of recurrent orbits is also proved. For the feedback control equation, these conditions are reduced to a form which can be easily verified in practice.