THE 2ND LOWEST EXTREMAL INVARIANT MEASURE OF THE CONTACT PROCESS

We study the ergodic behavior of the contact process on infinite conne cted graphs of bounded degree. We show that the fundamental notion of complete convergence is not as well behaved as it was thought to be. I n particular there are graphs for which complete convergence holds in any number of separated intervals of values of the infection parameter and fails for the other values of this parameter. We then introduce a basic invariant probability measure related to the recurrence propert ies of the process, and an associated notion of convergence that we ca ll ''partial convergence.'' This notion is shown to be better behaved than complete convergence, and to hold;in certain cases in which compl ete convergence fails. Relations between partial and complete converge nce are presented, as well as tools to verify when these properties ho ld. For homogeneous graphs we show that whenever recurrence takes plac e (i.e., whenever local survival occurs) there are exactly two extrema l invariant measures.