Efficient computations of Julia sets and their fractal dimension
The computation of the fractal dimension is straightforward using the box-counting method. However, this approach may require very long computation times. If the Julia set is the connected common boundary of two or more basins of attraction, then a recursive version of the box- counting method can be made storage- and time-efficient. The method is also suitable for the computation of the Julia sets. We apply the method to verify a result of D. Ruelle regarding the dimension of Julia sets of \(R(z)=z^ 2+c\) for small \(c\in {\mathbb{C}}\), to Newton's method for complex polynomials of degree 3 and to a sequence of Julia sets from the renormalization transformation for hierarchical lattices. We also discuss the computation of Julia sets and their information dimension by the inverse iteration method. In all examples tested we find that the information dimension is less than the fractal dimension.
scientific article; zbMATH DE number 4009373