Let \(M\) be a closed orientable surface of genus \(g\geq 2\) and \(f\) a diffeomorphism of \(M\). A classical result of Thurston states the existence of algebraic integers \( 1 \leq \lambda_1 < \dots < \lambda_K\) such that for any simple closed curve \(\alpha\) on \(M\), there is a \(\lambda_i\) such that \[ \lim_{n \rightarrow \infty} l_\rho (f^n \alpha)^{1/n} = \lambda_i \] where \(\rho\) is any Riemannian metric on \(M\) and \(l_\rho\) is the infimum of the length of curves isotopic to \(\alpha\). Moreover, \(f\) is pseudo-Anosov if and only if \(K=1\) and \(\lambda_1 >1\). The main purpose of the paper is to extend this result in two directions.
The first extension concerns random products of homeomorphisms. More precisely, for any integrable ergodic cocycle \(f_n = g_n g_{n-1} \cdots g_1\) of orientation-preserving homeomorphisms of \(M\), there exists almost surely a constant \(\lambda \geq 1\) and a measured foliation \(\mu\) such that for any simple closed curve \(\alpha\) such that \(i (\mu , \alpha) > 0\), and any Riemannian metric \(\rho\) : \[ \lim_{n \rightarrow \infty} l_\rho (f_n \alpha)^{1/n} = \lambda. \]
Moreover, given certain conditions of independence and support on the cocycle \(f_n\), one can obtain that \(\lambda > 1\). Heuristically, it means that random walk trajectories eventually look pseudo-Anosov from the perspective of Thurston’s result.
A second extension concerns holomorphic self-maps of Teichmüller space \(\mathcal{T}(M)\). For such a map \(f\) and a point \(x\) in Teichmüller space, there exists a number \(\lambda \geq 1\) such that the extremal length of any isotopy class of curve \(\beta\) on the complex structure \(f^n x\) is bounded below by \(\lambda^n\) times a constant.
It is well known that the mapping class group is isomorphic to the complex automorphism group of \(\mathcal{T}(M)\). So the previous result leads to a weak generalization of the Nielsen-Thurston classification of mapping classes to general holomorphic self-maps of Teichmüller spaces.