Issues of Chaos and Recurrence in Infinite Dimensions
arXiv:0911.2762
Abstract
Various issues with regard to chaos and recurrence in infinite dimensions are discussed. The doctrine we are trying to derive is that Sobolev spaces over bounded spatial domains do host chaos and recurrence, while Sobolev spaces over unbounded spatial domains are lack of chaos and recurrence. Local Sobolev spaces over unbounded spatial domains can host chaos and are natural phase spaces e.g. for fluid problems, but are very challenging to study.
Relevant manuscripts arXiv:0707.4458 and arXiv:0707.4456 [The Poincaré Recurrence Problem of Inviscid Incompressible Fluids, Asian J. Math, vol.13, no.1, 7-14, (2009). A Recurrence Theorem on the Solutions to the 2D Euler Equation Asian J. Math, vol.13, no.1, 1-6, (2009)]