Change of Velocity in Dynamical Systems

Abstract

In this work we study the properties of topological dynamical systems under a positive continuous change of velocity. In §l we define a flow obtained from a flow by a positive continuous change of velocity. We then prove that the time change flow is reversible so that we can recover the original flow. In §2 we define, following Kirillov, the first cohomology group of a dynamical system. A time change flow is then seen to be related to this first cohomology group. We now prove that there exists a group homomorphism between the first Cech cohomology group with integer coefficients and the first cohomology group of a compact dynamical system with coefficients in the reals. Winding numbers, due to Sol Schwartzman, are introduced and are shown to have an equivalent interpretation in terms of the first cohomology of a compact dynamical system. In §3 we show there is a natural invariant measure of a time change system in terms of the invariant measure of the original compact dynamical system. We now prove that ergodicity and unique ergodicity are preserved under a positive continuous change of velocity. Finally we relate the winding numbers of a time change system to the winding numbers of the original system, and show that under certain conditions they are invariant. In §4, we show that a compact dynamical system admits a Global Cross-Section if and only there exists an Eigen function, with non-zero eigenvalue, of a time change system. Lastly we show that, under certain conditions, a non-zero winding number is an eigenvalue associated to an eigen function of a time change dynamical system. In §5 we show that it is possible to eliminate eigen functions with non-zero eigenvalue under a positive continuous change of velocity of a compact dynamical system, if there exists at least one orbit homeomorphic to the real numbers : if, in addition, the original dynamical is ergodic we prove that weak-mixing is not invariant under a change velocity