H. van den Bedem and N. Chernov
EXPANDING MAPS OF AN INTERVAL WITH HOLES
(50K, LATeX)
ABSTRACT. We study a class of open chaotic dynamical systems.
Consider an expanding map of an interval from which a few small open
subintervals are removed (thus creating ``holes''). Almost every
point of the original interval then eventually escapes through the
holes, so there can be no absolutely continuous invariant
measures. We construct a so called conditionally invariant measure
that is equivalent to the Lebesgue measure. Our measure is unique
and naturally generates another measure, which is singular but
invariant. By this, we generalize early results by Pianigiani,
Yorke, Collet, Martinez and Schmidt, who studied similar maps
under an additional Markov assumption. We do not assume any Markov
property here and use ``bounded variation'' techniques rather than
Markov coding.