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Weakly chaotic non-linear maps with marginal fixed points have an infiniteinvariant measure. Time averages of integrable and non-integrable observablesremain random even in the long time limit. Temporal averages of integrableobservables are described by the Aaronson-Darling-Kac theorem. We find thedistribution of time averages of non-integrable observables, for example thetime average position of the particle. We show how this distribution is relatedto the infinite invariant density. We establish four identities betweenamplitude ratios controlling the statistics of the problem.

Infinite Invariant Density Determines Statistics of Time Averages for Weak Chaos