Rate of convergence in the disjunctive chaos game algorithm

Area of Science:

• Fractals and Dynamical Systems
• Computational Mathematics

Background:

• Iterated Function Systems (IFS) are fundamental to fractal geometry.
• The chaos game is a common algorithm for generating fractal attractors.

Purpose of the Study:

• To analyze the convergence rate of the chaos game algorithm.
• To understand the influence of symbolic sequences on convergence speed.

Main Methods:

• Studying the convergence rate of the chaos game algorithm.
• Analyzing the statistical properties of symbolic sequences driving the iteration.
• Estimating convergence exponents using box dimensions and driver entropy.

Main Results:

• Convergence is generally exponential, similar to Picard iterates.
• The symbolic sequence must exhibit specific statistical properties, like the Champernowne sequence.
• Convergence exponents are bounded by attractor dimensions and driver entropy.
• Generic drivers lead to variable convergence speeds, from arbitrarily slow to arbitrarily fast.

Conclusions:

• The chaos game's convergence rate is sensitive to the driving symbolic sequence's statistical behavior.
• Understanding these properties is crucial for efficient fractal attractor recovery.
• The interplay between attractor geometry and driver statistics dictates convergence characteristics.