Heat equation and ergodic theorems for Riemann surface laminations

Oseledec multiplicative ergodic theorem for laminations

“…However, she only deals with the following context: (X, L ) is a compact C 2 -smooth lamination endowed with a transversally continuous Riemannian metric g. In this context it is known (see, for instance, [5,3]) that (i) being valid for all t ∈ R + is equivalent to (ii), that is, weakly harmonic measures, harmonic measures in our sense and harmonic measures in the sense of Garnett are all equivalent. On the other hand, the same equivalence holds when (X, L ) is the regular part of a compact foliation by Riemann surfaces with linearizable singularities (see [12]). It is worthy noting here that, for every (X, L , g) satisfying Hypothesis (H1), we have that…”

“…In this context, the holonomy of leaves is closely related to the uniformizations of leaves and their Poincaré metric. This subject has received a lot of attention in the recent years (see, for example, the works by Candel [2], Candel-Gómez Mont [6], Dinh-Nguyen-Sibony [12,13,14], Fornaess-Sibony [16,17,18], Neto [28] etc). We also hope that the results of this Memoir may find applications in the dynamics of moduli spaces and in the geometric dynamics of laminations and foliations.…”

“…Such a foliation often admits a canonical Riemannian metric, namely, the Poincaré metric. However, this metric is transvesally measurable, it is continuous only in some good cases (see Dinh-Nguyen-Sibony [12], Fornaess-Sibony [17] etc). Our main examples of cocycles are the holonomy cocycles (or their tensor powers) of such foliations.…”

Singular holomorphic foliations by curves I: integrability of holonomy cocycle in dimension 2

“…The second main step is then devoted to the proof of Theorem 1.4, i.e., inequality (1.1). The main difficulty is that known estimates (see, for example, [10]) on the behavior of T near linearizable singularities, only give a weaker inequality…”

“…Roughly speaking, this step quantifies the expansion speed of the hololomy cocycle in terms of the ambient metric g X when one travels along unit-speed geodesic rays. The main ingredients are in our joint-works with Dinh and Sibony in [10,11,12]. The second main step is then devoted to the proof of Theorem 1.4, i.e., inequality (1.1).…”