Relative rotation number for stochastic systems: dynamical and topological applications

Abstract

We introduce a concept of relative rotation number to unify many different approaches of rotation number in non-linear dynamical systems. We present an ergodic result of existence a.s. for stochastic systems. In higher dimension, we show that the natural idea of projecting into a plane does work well a.s. for any plane (different from deterministic systems where projections may be degenerate). A number of further properties (invariance by homotopy and by conjugacy) and applications are presented.

Keywords:

• stochastic systems
• rotation number
• relative rotation
• invariance
• conjugacy
• homotopy

AMS 2000 Subject Classifications:

• 37E45
• 93E03
• 37C40 (341C40)

Acknowledgements

Research by Pedro J. Catuogno was partially supported by FAPESP, grant no. 02/10246-2. Research by Diego Sebastian Ledesma was supported by FAPESP, grant no. 04/13758-0. Research by Paulo R.C. Ruffino was partially supported by CNPq, grant no. 301112/2003-7 and FAPESP, grant no. 02/10246-2.