Inspired by the recent theory of Entropy-Transport problems and $\mathbf{D}$-distance Sturm on normalised metric measure spaces, we define a new class complete separable distances between spaces possibly different total mass. We provide several explicit examples such distances, where prominent role is played geodesic based Hellinger-Kantorovich distance. Moreover, discuss some limiting cases th...

We propose a method for testing homogeneity in three dimensional spatial distributions using Renyi entropy. apply the proposed to data from cosmological N-body simulations and Monte Carlo of homogeneous Poisson point process. show that can effectively characterize inhomogeneities identify any transition scale homogeneity, if present such distributions. The be used study cosmic future generation...

We prove that for a class of Zd shifts of finite type, d > 1, any invariant measure which is not a measure of maximal entropy can be perturbed a small amount in the weak* topology to an invariant measure of higher entropy. Namely, there are no invariant measures which are strictly local maxima for the entropy function.