FRACTAL ORACLE NUMBERS

Abstract

Consider orbits [Formula: see text] of the fractal iterator [Formula: see text], [Formula: see text], that start at initial points [Formula: see text], where [Formula: see text] is the set of all rational complex numbers (their real and imaginary parts are rational) and [Formula: see text] consists of all such [Formula: see text] whose complexity does not exceed some complexity parameter value [Formula: see text] (the complexity of [Formula: see text] is defined as the number of bits that suffice to describe the real and imaginary parts of [Formula: see text] in lowest form). The set [Formula: see text] is a bounded-complexity approximation of the filled Julia set [Formula: see text]. We present a new perspective on fractals based on an analogy with Chaitin’s algorithmic information theory, where a rational complex number [Formula: see text] is the analog of a program [Formula: see text], an iterator [Formula: see text] is analogous to a universal Turing machine [Formula: see text] which executes program [Formula: see text], and an unbounded orbit [Formula: see text] is analogous to an execution of a program [Formula: see text] on [Formula: see text] that halts. We define a real number [Formula: see text] which resembles Chaitin’s [Formula: see text] number, where, instead of being based on all programs [Formula: see text] whose execution on [Formula: see text] halts, it is based on all rational complex numbers [Formula: see text] whose orbits under [Formula: see text] are unbounded. Hence, similar to Chaitin’s [Formula: see text] number, [Formula: see text] acts as a theoretical limit or a “fractal oracle number” that provides an arbitrarily accurate complexity-based approximation of the filled Julia set [Formula: see text]. We present a procedure that, when given [Formula: see text] and [Formula: see text], it uses [Formula: see text] to generate [Formula: see text]. Several numerical examples of sets that estimate [Formula: see text] are presented.