Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure

Abstract

We consider Birkhoff averages of an observable $\phi$ along orbits of a continuous map $f:X \rightarrow X$ with respect to a non-invariant measure $m$. In the simple case where the averages converge $m$-almost everywhere, one may discuss the distribution of values of the average in a natural way. We extend this analysis to the case where convergence does not hold $m$-almost everywhere. The case that the averages converge $m$-almost everywhere is shown to be related to the recently defined notion of "predictable" behavior, which is a condition on the existence of pointwise asymptotic measures (SRB measures). A heteroclinic attractor is an example of a system which is not predictable. We define a more general notion called "statistically predictable" behavior which is weaker than predictability, but is strong enough to allow meaningful statistical properties, i.e. distribution of Birkhoff averages, to be analyzed. Statistical predictability is shown to imply the existence of an asymptotic measure, but not vice versa. We investigate the relationship between the various notions of asymptotic measures and distributions of Birkhoff average. Analysis of the heteroclinic attractor is used to illustrate the applicability of the concepts.