Discretizing randomly perturbed circle maps

Abstract

Let f be a C2 expanding map of the circle and = d m be the absolutely continuous invariant measure for this system. We formulate a model for a discrete approximation for by perturbing the measure and then discretizing it. We show that whenever the distance between lattice points of the discretization decays polynomially (but not linearly) in the perturbation parameter , then as tends to 0, the discretized density converges to in a suitable Hölder norm. Namely, if < 1+c for some c > 0, then || , ||(c) 0 as 0, where , is the discretized density. We also show that the rate of convergence is + c. This strengthens the work of Kifer, who shows (in a more general setting), weak convergence of the discretized perturbed measure to .