The number of infinite clusters in dynamical percolation

Dynamical percolation is a Markov process on the space of subgraphs of a given graph, that has the usual percolation measure as its stationary distribution. In previous work with O. Haggstrom, we found conditions for existence of infinite clusters at exceptional times. Here we show that for Z(d), with p > p(c), a.s. simultaneously for all times there is a unique infinite cluster, and the density of this cluster is theta(p). For dynamical percolation on a general tree Gamma, we show that for p > p(c), a.s. there are infinitely many infinite clusters at all times. At the critical value p = p(c), the number of infinite clusters may vary, and exhibits surprisingly rich behaviour. For spherically symmetric trees, we find the Hausdorff dimension of the set T-k Of times where the number of infinite clusters is k, and obtain sharp capacity criteria for a given time set to intersect T-k. The proof of this capacity criterion is based on a new kernel truncation technique.