Entropic Measure on Multidimensional Spaces

Abstract

We construct the entropic measure on compact manifolds of any dimension. It is defined as the push forward of the Dirichlet process (a random probability measure, well-known to exist on spaces of any dimension) under the conjugation map (formula presented) This conjugation map is a continuous involution. It can be regarded as the canonical extension to higher-dimensional spaces of a map between probability measures on 1-dimensional spaces characterized by the fact that the distribution functions of μ and C(μ) are inverse to each other. We also present a heuristic interpretation of the entropic measure as (formula presented).