For a given metric measure space ( X , d, μ ) we consider finite samples of points, calculate the matrix distances between them and then reconstruct points in some finite-dimensional using multidimensional scaling (MDS) algorithm with this distance as an input. We show that procedure gives natural limit number grows to infinity density approaches μ. This can be viewed “infinite MDS” embedding o...

We define a class of continuous graded graphs similar to the graph Gelfand–Tsetlin patterns, and describe set all ergodic central measures discrete type on path spaces such graphs. The main observation is that an measure subgraph Pascal-type can often be obtained as restriction standard Bernoulli space subgraph. This dramatically changes approach finding also graphs, famous Young graph. simples...

We show that if an infinite measure preserving system is well approximated on most of the phase space by a satisfying local limit theorem, then original enjoys mixing with respect to global observables, is, observables which admit volume average. The systems our conditions include Lorentz gas Coulomb potential, Galton board, and piecewise smooth Fermi-Ulam pingpongs.

This paper describes a certain part of Leonard Gross’s work in infinite-dimensional analysis, connected to the Gross Ergodicity Theorem. I then look at way in which Gross’s work helped to create a new subject within (mostly) finite-dimensional analysis, a subject which may be called “harmonic analysis with respect to heat kernel measure.” This subject transfers to Lie groups certain constructio...