[Jointly with Kyoto Dynamical Systems Seminar] Random maps: existence and approximation of invariant measures in one and higher dimensions

2017/12/22 Fri 15:00 - 16:30
Room 552, Building No.3
Shafiqul Islam
University of Prince Edward Island

A random map $T$ on a set $X$ is a dynamical system consisting a number of transformations from $X$ into itself, where the process switches from one map to another according to fixed probabilities or, more generally, position dependent probabilities. The existence and properties of invariant measures of random maps reflect their long time behavior and play an important role in understanding their statistical properties and chaotic nature. In this talk, first, we present the Frobenius-Perron operator as one of the key tools for the existence of absolutely continuous invariant measure (acim) for random maps. We present some result(s) on the existence of acims. Then, we present an open problem for the existence of infinite absolutely continuous invariant measure (acim) for random maps. Moreover, we talk about random maps in higher dimensions. In particular, we present random maps of piecewise real analytic expanding maps on the plane ($\mathbb{R}^2$). Finally, we present finite approximation(s) of the Frobenious-Perron operator and numerical method(s) for the approximation of acims for random maps.