Varia

Abstract

In this chapter, we collect different results related to some topics (like entropy or harmonic measures) discussed before. In the first section, we follow beautiful ideas due to Attie and Hurder [AH] who have shown that geometry of Riemannian manifolds quasi-isometric to leaves on compact foliated manifolds cannot be very chaotic in the sense that, on such a manifold, the maximal number of non-quasiisometric pieces of a given radius grows at most exponentially as a function of the radius. In Section 2, we introduce, after Inaba and Tsuchiya [IT], the notion of expansivity for pseudogroups and foliations and prove that expansive foliations of codimension 1 have strictly positive entropy and, therefore, contain some resilient leaves. In Section 3, we show how to calculate entropy of a group in terms of separated pseudo-orbits. This should provide a computational method of calculation (or, estimation) of entropies of groups, pseudogroups and foliations. Finally, Section 4 is devoted to the study of topology of generic, with respect to any harmonic measure, leaves. Following Ghys [Gh3] we show that there are just six topological types of such leaves in dimension 2.