FRACTAL ORACLE NUMBERS^*

Consider orbits O(z, κ) of the fractal iterator f_κ(z):= z² + κ, κ ∈ C, that start at initial points z ∈ K̂_κ⁽m⁾ ⊂ Ĉ, where Ĉ is the set of all rational complex numbers (their real and imaginary parts are rational) and K̂_κ⁽m⁾ consists of all such z whose complexity does not exceed some complexity parameter value m (the complexity of z is defined as the number of bits that suffice to describe the real and imaginary parts of z in lowest form). The set K̂_κ⁽m⁾ is a bounded-complexity approximation of the filled Julia set K_κ. We present a new perspective on fractals based on an analogy with Chaitin’s algorithmic information theory, where a rational complex number z is the analog of a program p, an iterator f_κ is analogous to a universal Turing machine U which executes program p, and an unbounded orbit O(z, κ) is analogous to an execution of a program p on U that halts. We define a real number Υ_κ which resembles Chaitin’s Ω number, where, instead of being based on all programs p whose execution on U halts, it is based on all rational complex numbers z whose orbits under f_κ are unbounded. Hence, similar to Chaitin’s Ω number, Υ_κ acts as a theoretical limit or a “fractal oracle number” that provides an arbitrarily accurate complexity-based approximation of the filled Julia set K_κ. We present a procedure that, when given m and κ, it uses Υ_κ to generate K̂_κ⁽m⁾. Several numerical examples of sets that estimate K̂_κ⁽m⁾ are presented.