Almost sure rotation number of piecewise affine endomorphisms of the circle (Q1865816)

The main result of the paper is the following theorem. Theorem C: A generic \(C^1\) vector field on a closed manifold \(M\) (where the set of \(C^1\) vector field spaces endowed with \(C^1\) topology is considered as a Baire topological space) has either infinitely many homoclinic classes, or else a finite collection of attractors with basins that form an open-dense set, and a finite collection of repellers with basins forming an open-dense set. It gives an approach to a conjecture of Palis. It is proved also that there exists a locally residual subset of \(C^1\) vector fields on a 5-manifold having finitely many attractors and repellers but infinitely many homoclinic classes.