Abstract:
Dynamical systems with holes model systems in which mass or energy is
allowed to escape over time and have attracted much attention over the
last ten years.  Typically, one starts with a closed system and
declares a subset of the phase space to be the 'hole,' essentially an
absorbing set. To date, most published works focus on systems in which
the rate of mixing, and thus the rate of escape, are exponential. This
talk will begin by reviewing some known results in the exponential
case and then proceed to investigate a class of polynomially mixing
systems with holes which exhibit qualitatively different bahavior,
which can be characterized as a loss of stability from the point of
view of the absolutely continuous invariant measure for the closed
system.  We will also present a more general result which indicates
that this loss of stability occurs typically for a subexponentially
mixing system.  We will then try to regain a version of stability by
looking at a sequence of measures supported on the survivor set of the
open system.  This will be a two-hour talk:  the first hour will
present a brief review and statement of recent results; the second
hour will present proofs.  Much of this is joint work with Bastien
Fernandez, CNRS.