We present a method of approximating the unique invariant measure and associated Lyapunov exponents of a random dynamical system deened by the iid composition of a family of maps in situations where only one Lyapunov exponent is observed. As a corollary, our construction also provides a method of estimating the top Lyapunov exponent of an iid random matrix product. We develop rigorous numerical...

We study a unitary version of the one-dimensional Anderson model, given by a five diagonal deterministic unitary operator multiplicatively perturbed by a random phase matrix. We fully characterize positivity and vanishing of the Lyapunov exponent for this model throughout the spectrum and for arbitrary distributions of the random phases. This includes Bernoulli distributions, where in certain c...

This work puts forward an extended definition of vector fractional Brownian motion (fBm) using a distribution theoretic formulation in the spirit of Gel’fand and Vilenkin’s stochastic analysis. We introduce random vector fields that share the statistical invariances of standard vector fBm (self-similarity and rotation invariance) but which, in contrast, have dependent vector components in the g...

We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g = 1. We also show when g = 1 or g = 2 that our bounds are best possible.

For prescribed values of a function and its partial derivatives of orders 1 and 2 at the vertices of a square, we fit an interpolating surface. We investigate two families of solutions provided by two Hermite subdivision schemes, denoted HD2 and HR2. Both schemes depend on 2 matrix parameters, a square matrix of order 2 and a square matrix of order 3. We exhibit the masks of both schemes. We co...

We present a fast, simple matrix method of computing the unique invariant measure and associated Lyapunov exponents of a nonlinear iterated function system. Analytic bounds for the error in our approximate invariant measure (in terms of the Hutchinson metric) are provided, while convergence of the Lyapunov exponent estimates to the true value is assured. As a special case, we are able to rigoro...

For prescribed values of a function and its partial derivatives of orders 1 and 2 at the vertices of a square, we fit an interpolating surface. We investigate two families of solutions provided by two Hermite subdivision schemes, denoted HD2 and HR2. Both schemes depend on 2 matrix parameters, a square matrix of order 2 and a square matrix of order 3. We exhibit the masks of both schemes. We co...