Information, Geometry, and Physics Seminar

An m-rectifiable measure is, roughly speaking, supported by an m-dimensional manifold, and a stratified measure is a convex combination of rectiffiable measures, possibly in different dimensions. Stratified measures generalize discrete-continuous mixtures and have, in general, a nontrivial singular continuous part. We shall study a set of typical realizations of n independent trials of an stratified measure. This set is also stratified and the dimensions of the strata concentrate around the expected value (that either coincides or is conjectured to coincide with Renyi's information dimension of the measure). The entropy of the stratified measure quantifies the exponential rate of growth of the typical set; it also verifies a chain rule whose conditional term bounds the rate of growth of typical realizations in each typical stratum.