BOUNDED COMPLEXITY APPROXIMATION OF FRACTAL SETS

Filled Julia sets K_k are fractals that consist of initial points of orbits that remain bounded under the application of an iterator map f_k, for instance quadratic maps z² + k. No known algorithm can determine, based on k alone, if an orbit that starts at z remains bounded. Hence in practice, to visualize such sets they are approximated using an escape time heuristic rule which approximates a dynamical orbit as being bounded if it remains so for some large but finite amount of time. The current paper considers a procedure that reproduces exactly a filled Julia set K_k, where k is a rational complex number, over a grid of arbitrary resolution, based only on k and an oracle number that depends on the complexity of elements of the set K_k over the grid. The procedure outputs a finite set (Formula presented.) of rational complex numbers in K_k whose complexity is bounded from above by a parameter value m. A sufficient condition on m as a function of a given positive integer parameter N is obtained that ensures that (Formula presented.) is an exact approximation (reproduction) of K_k over an N × N grid. An interesting consequence is that for arbitrarily large N, given that k is known, the cummulative information about the complexity of all rational z in the complement of K_k determines the asymptotic dynamics of their corresponding orbits.