Piecewise Convex Deterministic Dynamical Systems and Weakly Convex Random Dynamical Systems and Their Invariant Measures

Abstract

Absolutely continuous invariant measures of deterministic dynamical systems and random dynamical systems respectively are studied via a general spline maximum entropy optimization method. In the first part of this paper, we consider piecewise convex deterministic dynamical systems (maps) $$\tau : [0, 1]\rightarrow [0, 1]$$ and we study their absolutely continuous invariant measures. We assume that the deterministic piecewise convex transformation $$\tau $$ has a unique absolutely continuous invariant measure (acim) $$\mu ^*$$ with density $$f^*.$$ We present a general spline maximum entropy optimization method for the approximation of $$f^*.$$ The proof of convergence of our general spline maximum entropy optimization method is presented. A numerical example is presented for the general spline (linear, quadratic and cubic respectively) maximum entropy numerical scheme for the approximation of $$f^*.$$ In the second part of this paper, we generalize above results for weakly convex position dependent random map $$T=\{\tau _1(x),\tau _2(x),\ldots , \tau _K(x); p_1(x),p_2(x),\ldots ,p_K(x)\}$$ on $$I=[0, 1],$$ where $$\tau _k: [0, 1] \rightarrow [0, 1]), k=1, 2, \dots , K$$ is a piecewise convex map and $$\{p_1(x), p_2(x),\ldots ,p_K(x) \}$$ is a set of position dependent probabilities on [0, 1]. We assume that T has a unique acim $$\nu ^*$$ with density $$h^*.$$ We present a general spline maximum entropy optimization method for the approximation of $$h^*.$$ The proof of convergence of our numerical schemes is presented. Also, we present a numerical example of the general spline maximum entropy method for the approximation of $$h^*.$$