event
Wednesday 22 Jun 2011: Invariant measures with bounded variation densities for Piecewise Area Preserving Maps
Yiwei Zhang -
TBC 15:00-16:00
We investigate the properties of absolutely continuous invariant
probability measures (ACIPs) for piecewise area preserving maps (PAPs) on $\mathbb{R}^d$. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. In particular for PWIs, we use functional approach to explore the relationship between topological transitivity and uniqueness of ACIPs, especially those measures with bounded variation densities. This ``partially'' answers one of the fundamental questions posed in \cite{Goetz03} - determine all invariant non-atomic probability Borel measures in piecewise rotation. When reducing to interval exchange transformations (IETs), we demonstrate that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have a very irregular density (namely of unbounded variation and discontinuous everywhere) and intermingle with each other.
