Bounded complexity approximation of fractal sets

Filled Julia sets $ K_{\kappa} $ are fractals that consist of initial points of orbits that remain bounded under the application of an iterator map $ f_{\kappa} $, for instance quadratic maps $ z^{2}+\kappa $. No known algorithm can determine, based on $ \kappa $ 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_{\kappa} $, where $ \kappa $ is a rational complex number, over a grid of arbitrary resolution, based only on $ \kappa $ and an oracle number that depends on the complexity of elements of the set $ K_{\kappa} $ over the grid. The procedure outputs a finite set $ \hat{K}_{\kappa}^{(m)} $ of rational complex numbers in $ K_{\kappa} $ 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 $ \hat{K}_{\kappa}^{(m)} $ is an exact approximation (reproduction) of $ K_{\kappa} $ over an $ N\times N $ grid. An interesting consequence is that for arbitrarily large $ N $, given that $ \kappa $ is known, the cummulative information about the complexity of all rational $ z $ in the complement of $ K_{\kappa} $ determines the asymptotic dynamics of their corresponding orbits.