  We study how electron-electron interactions renormalize tunneling into a
Luttinger liquid beyond the lowest order of perturbation in the tunneling
amplitude. We find that the conventional fixed point has a finite basin of
attraction only in the point contact model, but a finite size of the contact
makes it generically unstable to the tunneling-induced break up of the liquid
into two independent parts. In the course of renormalization to the
nonperturbative-in-tunneling fixed point, the tunneling conductance may show a
nonmonotonic behavior with temperature or bias voltage.
