Ulam's method for random interval maps

Abstract

We consider the approximation of absolutely continuous invariant measures (ACIMs) of systems defined by random compositions of piecewise monotonic transformations. Convergence of Ulam's finite approximation scheme in the case of a single transformation was dealt with by Li (1976 J. Approx. Theory 17 177-86). We extend Ulam's construction to the situation where a family of piecewise monotonic transformations are composed according to either an iid or Markov law, and prove an analogous convergence result. In addition, we obtain a convergence rate for our approximations to the unique ACIM, and provide rigorous bounds for the L 1 error of the Ulam approximation.

Extracting Dynamical Behavior via Markov Models

“…Recent results (often under additional "ontoness" assumptions) have focused on expllcit error bounds for the difference 11J.l-J.lnll£1; [17,35,49]. For higher-dimensional uniformly expanding systems, very roughly speaking, the papers of Boyarsky and Gora [27] and Ding [12] mirror those of [43] and [44].…”

“…Under some additional conditions, it is shown in [17] that (i) the random system (either Ll.D. or Markov) possesses a unique bounded invariant density, and (ii) that the Ulam estimates J-Ln converge to the physical measure J-L (which has a bounded density).…”

“…and Markov cases, the matrices Pn and Sn may be thought of as finite-dimensional projections of an "averaged" PerronFrobenius operator; see [17] for details. With either of these two matrices, one may apply the methods described in Section 12.3 to estimate invariant objects such as invariant measures, invariant sets, Lyapunov exponents, and recurrence times.…”

A closing scheme for finding almost-invariant sets in open dynamical systems

“…A rigorous justification for the approximation µ n goes back to Li [35] who considered expanding, piecewise C 2 maps T of the unit interval. Since then, similar results, extending the class of maps have been developed [24,21,25,38]. In the open setting, the approximation ν n has also been rigorously justified for expanding maps of the interval [1].…”

Generalized Cell Mapping for Randomly Perturbed Dynamical Systems

“…They analyze the dynamics path-wise, i.e., for one realization of the random process assuming an ergodic system. Convergence proofs for related methods have been given by Froyland [5], Hunt [10] and by Dellnitz and Junge [3].…”