A Matrix Contraction Process

We consider a stochastic process in which independent identically distributed random matrices are multiplied and where the Lyapunov exponent of the product is positive. We continue multiplying the random matrices as long as the norm, $\epsilon$, of the product is \emph{less} than unity. If the norm is greater than unity we reset the matrix to a multiple of the identity and then continue the multiplication. We address the problem of determining the probability density function of the norm, $P_\epsilon$. We argue that, in the limit as $\epsilon\to 0$, $P_\epsilon\sim (\ln (1/\epsilon))^\mu \epsilon^\gamma$, where $\mu$ and $\gamma$ are two real parameters. Our motivation for analysing this \emph{matrix contraction process} is that it serves as a model for describing the fine-structure of strange attractors, where a dense concentration of trajectories results from the differential of the flow being contracting in some region. We exhibit a matrix-product model for the differential of the flow in a random velocity field, and show that there is a phase transition, with the parameter $\mu$ changing abruptly from $\mu=0$ to $\mu=-\frac{3}{2}$ as a parameter of the flow field model is varied.