Invariant Measures in Random Dynamical Systems

How random dynamics can generate fractal invariant measures.

Summary

This project studies how a simple random dynamical system can produce a fractal invariant measure. The simulation selects actions from a family of maps and visualizes the long-run distribution generated by repeated random composition.

Why Invariant Measures Matter

An invariant measure describes the statistical state that remains unchanged under the dynamics. In random systems, the individual trajectory may look irregular, but the distribution of many trajectories can converge toward a stable geometric pattern. That makes invariant measures a useful bridge between stochastic rules and visible structure.

Example

Consider a simple random dynamical system on a torus:

$$ {x_{n + 1}} = \left\{ \begin{array}{l} {f_0}({x_n}) = 4{x_n}\bmod \,1\\ {f_i}({x_n}) = \frac{x_n + i - 1}{16} \end{array} \right. $$

The index $i$ ranges from $1$ to $16$. At each step, the expansive map $f_0$ is chosen with probability $1/2$, while each contracting map $f_i$ is chosen with probability $1/32$.

For a symbolic sequence \(\omega = a_0a_1a_2...a_N\) with \(a_i=0,...,16\), define \(f_{\omega} = f_{a_N} \circ ... \circ f_{a_0}\). Starting from samples drawn uniformly on the square, the simulation tracks the distribution of \((f_{\omega}(x),f_{\omega}(y))\) over repeated random compositions.

Display

Scatter plot showing a fractal invariant measure in a random dynamical system
Each point represents $$(f_{\omega}(x),f_{\omega}(y))$$ for two random variables. The density of points reveals the invariant measure, while the diagonal line marks synchronization between the two initial states.

Takeaway

The example shows how randomness and contraction can cooperate rather than cancel one another: repeated stochastic choices produce a stable distribution whose support has a visible fractal geometry.