The Boundary of Periodic Iterated Function Systems

This Demonstration shows approximations of the boundary (clockwise) of some discrete families of iterated functional systems (IFS), , which can be thought of as the fractional part of numeration systems with a complex base, .

Change and to choose one of the periodic cases. The next slider "shown recurrence level" lets you choose the level of approximation and the rotation of the recurrence relation; for example, '' represents the iteration. These relations show that an edge with given previous direction (red) changes into the resulting sequence of edges. Ignoring order gives the matrix with dominant eigenvalue () and corresponding eigenvector, which gives the limit distribution of the number of edges. In the limit the figure becomes a fractal with the Hausdorff dimension .