Even the first iterate of a Markov operator is contracting in anL 2 norm

A weightedL ₂ norm is introduced in which Markov operators, e.g., associated with noisy maps, are contracting provided the kernel (i.e., the transitional distribution) is smooth enough. This results in strong relaxational properties of noisy maps. Similar to this norm, integral functionals appear useful when studying spatiotemporal chaos and random fields.