Entropic Measure on Multidimensional Spaces

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We construct the entropic measure Pβ on compact manifolds of any dimension. It is defined as the push forward of the Dirichlet process (another random probability measure, well-known to exist on spaces of any dimension) under the conjugation map:P(M)(M). 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 (μ) are inverse to each other. We also present an heuristic interpretation of the entropic measure as dPβ(μ)=1/Z(-β· Ent (μ|m))· dP⁰(μ).