  One goal of this paper is to prove that dynamical critical site percolation
on the planar triangular lattice has exceptional times at which percolation
occurs. In doing so, new quantitative noise sensitivity results for percolation
are obtained. The latter is based on a novel method for controlling the "level
k" Fourier coefficients via the construction of a randomized algorithm which
looks at random bits, outputs the value of a particular function but looks at
any fixed input bit with low probability. We also obtain upper and lower bounds
on the Hausdorff dimension of the set of percolating times. We then study the
problem of exceptional times for certain "k-arm" events on wedges and cones. As
a corollary of this analysis, we prove, among other things, that there are no
times at which there are two infinite "white" clusters, obtain an upper bound
on the Hausdorff dimension of the set of times at which there are both an
infinite white cluster and an infinite black cluster and prove that for
dynamical critical bond percolation on the square grid there are no exceptional
times at which three disjoint infinite clusters are present.
