Generic leaves

Abstract.

A remarkable theorem of E. Ghys asserts that, for any harmonic measure $\mu$ on a compact, foliated metric space, $\mu$-almost every leaf has 0, 1, 2 or a Cantor set of ends. In this paper, analogous results are proven for topologically almost all (i.e., residual families of) leaves. More precisely, if some leaf is totally recurrent, a residual family of leaves is totally recurrent with 1, 2 or a Cantor set of ends. A "local" version of this theorem asserts that, in general, topologically almost all leaves have 0, 1, 2 or a Cantor set of dense ends.