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A higher dimensional Markov switching position dependent random map is a random map where the probabilities of switching from one higher dimension transformation to another are the entries of a stochastic matrix and the entries of stochastic matrix are functions of positions. In this note, we prove sufficient conditions for the existence of absolutely continuous measures for a class of higher dimensional Markov switching position dependent random maps. Our result is a generalization of the result in [Bahsoun & Góra, 2005; Bahsoun et al., 2005].

Metastable dynamical systems were recently studied [González-Tokman et al., 2011] in the framework of one-dimensional piecewise expanding maps on two disjoint invariant sets, each possessing its own ergodic absolutely continuous invariant measure (acim). Under small deterministic perturbations, holes between the two disjoint systems are created, and the two ergodic systems merge into one. The long term dynamics of the newly formed metastable system is defined by the unique acim on the combined ergodic sets. The main result of [González-Tokman et al., 2011] proves that this combined acim can be approximated by a convex combination of the disjoint acims with weights depending on the ratio of the respective measures of the holes. In this note we present an entirely different approach to metastable systems. We consider two piecewise expanding maps: one is the original map, τ₁, defined on two disjoint invariant sets of ℝ^N and the other is a deterministically perturbed version of τ₁, τ₂, which allows passage between the two disjoint invariant sets of τ₁. We model this system by a position dependent random map based on τ₁ and τ₂, to which we associate position dependent probabilities that reflect the switching between the maps. A typical orbit spends a long time in one of the ergodic sets but eventually switches to the other. Such behavior can be attributed to physical holes as between adjoining billiard tables or more abstract situations where balls can "leap" from one table to the other. Using results for random maps, a result similar to the one-dimensional main result of [González-Tokman et al., 2011] is proved in N dimensions. We also consider holes in more than two invariant sets. A number of examples are presented.

Let T = {τ₁, τ₂, …, τ_K; p₁, p₂, …, p_K} be a position dependent random map on [0, 1], where {τ₁, τ₂, …, τ_K} is a collection of nonsingular maps on [0, 1] into [0, 1] and {p₁, p₂, …, p_K} is a collection of position dependent probabilities on [0, 1]. We assume that the random map T has a unique absolutely continuous invariant measure μ with density f*. Based on interpolation, a piecewise linear approximation method for f* is developed and a proof of convergence of the piecewise linear method is presented. A numerical example for a position dependent random map is presented.

We present a numerical method for the approximation of absolutely continuous invariant measures of one-dimensional random maps, based on the maximum entropy principle and piecewise linear moment functions. Numerical results are also presented to show the convergence of the algorithm.

Let $S$ be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map