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5. Quantum Fractals and IFS

Our algorithm is, in fact, a version of a nonlinear iterated function system (IFS).

Figure: The classical fractal: Sierpinski Triangle generated by an Iterated Function System.

Such algorithms are known to produce complex geometrical structures by repeated application of several non-commuting affine maps. The best known example is the Sierpinski triangle generated by random application of $3\times 3$ matrices $A[i], i=1,2,3$ to the vector:
\begin{displaymath} v_0=\pmatrix{x_0\cr y_0\cr 1} \end{displaymath} (21)

where $A[i]$ is given by
\begin{displaymath} A[i]=\pmatrix{0.5&0&ax_i\cr 0&0.5&ay_i\cr0&0&1} \end{displaymath} (22)

and $ax_1=1.0, ay_1=1.0, ax_2=1.0, ay_2=0.5, ax_3=0.5, ay_3=1.0.$ (Our $3\times 3$ matrices encode affine transformations - usually separated into a $2\times 2$ matrix and a translation vector.) At each step one of the three transformations $A[i], i=1,2,3$ is selected with probability $p[i]=1/3$. After each transformation the transformed vector is plotted on the $(x,y)$ plane. Theoretical papers on IFSs usually assume that the system is hyperbolic that is that each transformation is a contraction, i.e. the distances between points get smaller and smaller. It was shown in [17] that this assumption can be essentially relaxed when transformations are non-linear and act on a compact space - as is in the case of quantum fractals we are dealing with.
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Next: 6. Continuous part of Up: 4. Notes Previous: 4. Geometrical meaning of

2002-04-11