Infinite cascades of braids and smooth dynamical systems
The authors consider the linking between orbits in a compact invariant set of a nonsingular vector field defined on a region in \(\mathbb{R}^ 3\). They discuss and compare some concepts of linking numbers of orbits and prove results concerning the behaviour of the sequence of the so-called average linking numbers between successive periodic orbits in the cascade generated by a \(C^ 1\)-flow on the solid torus and, then, reformulate this result in terms of an orientation preserving \(C^ 1\)-diffeomorphism of the 2-disk. They also show that, given a special sequence \((\ell_ n)\) of rational numbers, there exists a \(C^ 1\)-diffeomorphism (with zero topological entropy) of the 2-disk realizing a cascade of periodic orbits with the prescribed period and \(\ell_ n\) as the average linking number between successive orbits.
scientific article; zbMATH DE number 527439