Models of eternal inflation predict a stochastic self-similar geometry of the
universe at very large scales and allow existence of points that never
thermalize. I explore the fractal geometry of the resulting spacetime, using
coordinate-independent quantities. The formalism of stochastic inflation can be
used to obtain the fractal dimension of the set of eternally inflating points
(the ``eternal fractal''). I also derive a nonlinear branching diffusion
equation describing global properties of the eternal set and the probability to
realize eternal inflation. I show gauge invariance of the condition for
presence of eternal inflation. Finally, I consider the question of whether all
thermalized regions merge into one connected domain. Fractal dimension of the
eternal set provides a (weak) sufficient condition for merging.Comment: Conversion to RevTeX4; minor changes; version accepted by Phys. Rev.
