# Spectral Theorems for Random Walks on Mapping Class Groups and Out $(\boldsymbol{F}_{\boldsymbol{N}})$

@article{Dahmani2015SpectralTF, title={Spectral Theorems for Random Walks on Mapping Class Groups and Out \$(\boldsymbol\{F\}\_\{\boldsymbol\{N\}\})\$}, author={François Dahmani and Camille Horbez}, journal={International Mathematics Research Notices}, year={2015}, volume={2018}, pages={2693-2744} }

We establish spectral theorems for random walks on mapping class groups of connected, closed, oriented, hyperbolic surfaces, and on $\text{Out}(F_N)$. In both cases, we relate the asymptotics of the stretching factor of the diffeomorphism/automorphism obtained at time $n$ of the random walk to the Lyapunov exponent of the walk, which gives the typical growth rate of the length of a curve -- or of a conjugacy class in $F_N$ -- under a random product of diffeomorphisms/automorphisms. In the…

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