The Gaussian multiplicative chaos is a relatively new universal object in probability that has many interesting geometric properties.The characteristic polynomial of many classes of random matrices is, in many cases conjecturally, one class of finite approximation to these random measures. Great progress has been made on showing the random matrices from specific ``circular ensembles" converge to the GMC. Likewise, some progress has been made for unitarily-invariant random matrices. We show some new partial progress in showing the ``Gaussian-beta ensemble'' has a GMC limit. This we do by using the representation of its characteristic polynomial as an entry in a product of independent random two-by-two matrices. For a point $z$ in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane. This is joint work with Gaultier Lambert.