Invariant Measure in Random Dynamical System

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 \((f_{\omega}(x),f_{\omega}(y))\).

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