Ergodic Theorems for Laminations and Foliations: Recent Results and Perspectives
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Ergodic Theorems for Laminations and Foliations: Recent Results and Perspectives ˆ ˆ 1,2 Viet-Anh Nguyen Dedicated to My Beloved Father Received: 10 December 2019 / Accepted: 21 March 2020 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract This report discusses recent results as well as new perspectives in the ergodic theory for Riemann surface laminations, with an emphasis on singular holomorphic foliations by curves. The central notions of these developments are leafwise Poincar´e metric, directed positive harmonic currents, multiplicative cocycles, and Lyapunov exponents. We deal with various ergodic theorems for such laminations: random and operator ergodic theorems, (geometric) Birkhoff ergodic theorems, Oseledec multiplicative ergodic theorem, and unique ergodicity theorems. Applications of these theorems are also given. In particular, we define and study the canonical Lyapunov exponents for a large family of singular holomorphic foliations on compact projective surfaces. Topological and algebro-geometric interpretations of these characteristic numbers are also treated. These results highlight the strong similarity as well as the fundamental differences between the ergodic theory of maps and that of Riemann surface laminations. Most of the results reported here are known. However, sufficient conditions for abstract heat diffusions to coincide with the leafwise heat diffusions (Section 5.2) are new ones. Keywords Riemann surface lamination · Singular holomorphic foliation · Leafwise Poincar´e metric · Positive harmonic currents · Multiplicative cocycles · Ergodic theorems · Lyapunov exponents Mathematics Subject Classification (2020) Primary 37A30 · 57R30 · Secondary 58J35 · 58J65 · 60J65 · 32J25
Lecture at the Annual Meeting 2019 of the Vietnam Institute for Advanced Study in Mathematics Viˆet-Anh Nguyˆen
[email protected] 1
Laboratoire de math´ematiques Paul Painlev´e, CNRS U.M.R. 8524, Universit´e de Lille, 59655, Villeneuve d’Ascq Cedex, France
2
Vietnam Institute for Advanced Study in Mathematics, 157 Chua Lang Street, Hanoi, Vietnam
V.-A. Nguyˆen
1 Introduction 1.1 Prelude The goal of these notes is to explain recent ergodic theorems for laminations by Riemann surfaces (without and with singularities), and particularly those for singular holomorphic foliations by curves. We make an emphasis on the analytic approach to the dynamical theory of laminations and foliations. This illustrates a prominent role of the theory of currents in the field. There is a natural correspondence between the dynamics of Riemann surface laminations and those of iterations of continuous maps. More specifically in the meromorphic category, this correspondence becomes a connection between the dynamics of singular holomorphic foliations in dimension k ≥ 2 and those of iterations of meromorphic maps in dimension k − 1. Ergodic theorems for measurable maps are by now well-understood, see, for instance, the monograph of Krengel [68]
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