Narrow Results

We study position dependent random maps on the unit interval with holes where the possible laws of motion are piecewise monotonic transformations. The main result of this note is proving the existence of absolutely continuous conditionally invariant measures.

We answer a question posed by Buzzi about the quasi-compactness of the Frobenius–Perron operator of random β-transformations for β∈(1,∞).

We present a rigorous convergence analysis of a linear spline Markov finite approximation method for computing stationary densities of random maps with position dependent probabilities, which consist of several chaotic maps. The whole analysis is based on a new Lasota–Yorke-type inequality for the Markov operator associated with the random map, which is better than the previous one in the literature and much simpler to obtain. We also present numerical results to support our theoretical analysis.

We prove the $L^1$-norm and bounded variation norm convergence of a piecewise linear least squares method for the computation of an invariant density of the Foias operator associated with a random map with position dependent probabilities. Then we estimate the convergence rate of this least squares method in the $L^1$-norm and the bounded variation norm, respectively. The numerical results, which demonstrate a higher order accuracy than the linear spline Markov method, support the theoretical analysis.

We study the dynamics of a new family of transformations. We find a general formula for the invariant density of any transformation in our family. The properties of the family allow us to prove that the invariant density function $f$ of random map constructed from our family maps $T=\{{\tau }_1,{\tau }_2,\dots ,{\tau }_n;p_1,p_2,\dots ,p_n\}$ is the combination $f=p_1{f}_1+p_2{f}_2+\cdots +p_n{f}_n,$ where $f_1,f_2,\dots ,f_n$ are the invariant density functions of ${\tau }_1,{\tau }_2,\dots ,{\tau }_n,$ respectively. We also consider another family of transformations, and prove that the invariant density for any transformation of the family is $f=1$.

A random map is a dynamical system consisting of a collection of maps which are selected randomly by means of fixed probabilities at each iteration. In this note, we consider absolutely continuous invariant measures of random maps with position dependent probabilities and prove that they are stable under small stochastic perturbations. This result depends on a new lemma which handles arbitrarily small extra partition elements that may arise from the perturbation of the random map. For perturbations satisfying additional conditions, we give precise estimates of the error in the invariant density.

A random map is a discrete-time dynamical system in which a transformation is randomly selected from a collection of transformations according to a probability function and applied to the process. In this note, we study random maps with position-dependent probabilities on ℝ. This means that the random map under consideration consists of transformations which are piecewise monotonic with countable number of branches from ℝ into itself and a probability function which is position dependent. We prove existence of absolutely continuous invariant probability measures and construct a method for approximating their densities. Explicit quantitative bound on the approximation error is given.