Invariant Measure in Random Dynamical System
What is the mechanism of fractals in nature?
Summary
Implemented a simulation to automatically select action sets that can lead to fractal invariant measure.
Example
Consider a simple random dynamical system on a torus
$$ {x_{n + 1}} = \left\{ \begin{array}{l} {f_0}({x_n}) = 4{x_n}\bmod \,1\\ {f_i}({x_n}) = \frac{x_n + i - 1}{16} \end{array} \right. $$
i is ranging from 1,…,16. And there is 1/2 probability to choose \(f_0\) and \(1/32\) probability to choose any \(f_i\).
Consider \(\omega = a_0a_1a_2...a_N, a_i=0,...,16\) is a symbolic sequence, and \(f_{\omega} = f_{a_N} \circ ... \circ f_{a_0}\).
For a set of sampling from the uniform distribution on square, consider the probability measure of \((f_{\omega}(x),f_{\omega}(y))\).
Display
An illustration of the invariant measure. Each scatter represents a point $$(f_{\omega}(x),f_{\omega}(y))$$ of two random variables. The density of the scatters
represents the invariant probability measure. The diagonal line represents the synchronization of two initial states.