  We investigate the percolation properties of a two-state (occupied - empty)
cellular automaton, where at each time step a cluster of occupied sites is
removed and the same number of randomly chosen empty sites are occupied again.
We find a finite region of critical behavior, formation of synchronized
stripes, additional phase transitions, as well as violation of the usual
finite-size scaling and hyperscaling relations, phenomena that are very
different from conventional percolation systems. We explain the mechanisms
behind all these phenomena using computer simulations and analytic arguments.
