Abstract: A filtration on a section ring is called submultiplicative if it respects the multiplicative structure of the section ring in the following way: the weight of a product of two sections is at least as big as the sum of the weights of the two sections. The most natural example is the filtration associated with a divisor; the weight of a holomorphic section is defined as the order of vanishing along the divisor. Study of the asymptotic properties of submultiplicative filtrations (as the number of sections with weights in a given interval) is a subject of several investigations, see Boucksom-Chen, Chen-Maclean, and it is related with K-stability, obstructing existence of constant scalar curvature K ̈ahler metrics. We will show, following Phong-Sturm, Ross-Witt Nystr ̈om and Berman-Boucksom-Jonsson, that to any submultiplicative filtration one can naturally associate a Mabuchi geodesic ray in the space of K ̈ahler metrics through the solution of Dirichlet problem for the Monge-Amp`ere operator on the test configuration associated with the filtration. Following Hisamoto, by relying on the results of previous lectures, we establish that the asymptotic properties of the filtrations are related with Mabuchi geometry at the infinity of the associated geodesic rays.