  We show that dynamical systems with $\phi$-mixing measures have local escape
rates which are exponential with rate $1$ at non-periodic points and equal to
the extremal index at periodic points. We apply this result to equilibrium
states on subshifts of finite type, expanding interval maps, Gibbs states on
conformal repellers and more generally to Young towers and by extension to all
systems that can be modeled by a Young tower.
