We consider dynamical systems on domains that are not invariant under the dynamics—for example, a system with a hole in the phase space—and raise issues regarding the meaning of escape rates and conditionally invariant measures. Equating observable events with sets of positive Lebesgue measure, we are led quickly to conditionally invariant measures that are absolutely continuous with respect to Lebesgue. Comparisons with SRB measures are inevitable, yet there are important differences. Via informal discussions and examples, this paper seeks to clarify the ideas involved. It includes also a brief review of known results and possible directions of further work in this developing subject.
Demers, Mark and Young, Lai-Sang, "Escape rates and conditionally invariant measures" (2006). Mathematics Faculty Publications. 42.
Mark Demers and Lai-Sang Young, "Escape rates and conditionally invariant measures," Nonlinearity, 19:2 (2006), 377-397.