Random walk boundaries: their entropies and connections with Hecke pairs

We present three papers in non-singular dynamics concerning boundaries of random walks on locally compact, second countable groups. One common theme is entropy. Paper II and III are concerned with boundary entropy spectra, while Paper I studies topological properties of entropy. In Paper II we moreover establish a technique to relate random walks on locally profinite groups to random walks on dense discrete subgroups, by the concept of Hecke pairs, which is also used in Paper III.In Paper I we introduce different perspectives and extensions of Furstenberg\u27s entropy and show semi-continuity and continuity results in these contexts. In particular we apply these to upper and lower limits of non-nested sequences of sigma-algebras in the sense of Kudo.Paper II relates certain random walks on locally profinite groups to random walks on dense discrete subgroups, using a Hecke subgroup, such that the Poisson boundary of the first becomes a boundary of the second one. If the Poisson boundaries of these two walks happen to coincide, then the Hecke subgroup in charge has to be amenable. For some random walks on lamplighter and solvable Baumslag-Solitar groups we obtain that their Poisson boundary is prime and the quasi-regular representation is reducible. Moreover, we find a group such that for any given summable sequence of positive numbers there is a random walk whose boundary entropy spectrum equals the subsum set of this sequence. In particular we obtain a boundary entropy spectrum which is a Cantor set and one which is an interval.In Paper III we study the boundary entropy spectra of finitely supported, generating random walks on a certain affine group, realizing them as finite subsum sets. We show that the averaged information function of a stationary probability measure does not change when passing to a non-singular, absolutely continuous sigma-finite measure and deduce an entropy formula