Lyapunov exponents, bifurcation currents and laminations in bifurcation loci

Geometric characterization of Lyapunov exponents for Riemann surface laminations

We characterize geometrically the Lyapunov exponents of a cocycle (of arbitrary rank) with respect to a harmonic current defined on a hyperbolic Riemann surface lamination. Our characterizations are formulated in terms of the expansion rates of the cocycle along geodesic rays.

内函数与层叠结构

Inner Functions and Laminations

本文利用黎曼曲面层叠结构上的测地流和环面流研究内函数的轨道计数问题，证明了有限李雅普诺夫指数内函数的原像计数渐近公式，并揭示几乎每条逆轨道的迭代具有渐近线性性质。

Oseledec multiplicative ergodic theorem for laminations

Given a n-dimensional lamination endowed with a Riemannian metric, we introduce the notion of a multiplicative cocycle of rank d, where n and d are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, we define the Lyapunov exponents of such a cocycle with respect to a harmonic probability measure directed by the lamination. We also prove an Oseledec multiplicative ergodic theorem in this context. This theorem implies the existence of an Oseledec decomposition almost everywhere w…

Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of CP(k)

We introduce a notion of stability for equilibrium measures in holomorphic families of endomorphisms of CP(k) and prove that it is equivalent to the stability of repelling cycles and equivalent to the existence of some measurable holomorphic motion of Julia sets which we call equilibrium lamination. We characterize the corresponding bifurcations by the strict subharmonicity of the sum of Lyapunov exponents or the instability of critical dynamics and analyze how repelling cycles may bifurcate. Our methods deeply exploit the properties of Lyapunov exponents and are based on ergodic theory and on…

Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton's Methods

We investigate i.i.d. random complex dynamical systems generated by probability measures on finite unions of the loci of holomorphic families of rational maps on the Riemann sphere. We show that under certain conditions on the families, for a generic system, (especially, for a generic random polynomial dynamical system,) for all but countably many initial values $z$ in the Riemann sphere, for almost every sequence of maps $γ=(γ_{1},γ_{2},\ldots )$, the Lyapunov exponent of $γ$ at $z$ is negative. Also, we show that for a generic system, for every initial value $z$ in the Riemann sphere, the or…