Products of random matrices: a dynamical point of view

Lucas Kaufmann (Department of Mathematics, National University of Singapore; and Center for Complex Geometry, Institute for Basic Science, Daejeon, South Korea)

Hao Wu (Department of Mathematics, National University of Singapore)

Abstract

We study products of random matrices in $\mathrm{SL}^2 (\mathbb{C})$ from the point of view of holomorphic dynamics. For non-elementary measures with finite first moment we obtain the exponential convergence towards the stationary measure in Sobolev norm. As a consequence we obtain the exponentially fast equidistribution of forward images of points towards the stationary measure. We also give a new proof of the Central Limit Theorem for the norm cocycle under a second moment condition, originally due to Benoist–Quint, and obtain some general regularity results for stationary measures.

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This work was supported by the NUS grants C-146-000-047-001, AcRF Tier 1 R- 146-000-248-114 and R-146-000-259-114.

Received 23 May 2019

Accepted 8 January 2020

Published 14 June 2021