Fractal Oracle Numbers

Consider orbits 𠒪(z,κ) of the fractal iterator fκ(z) := z2 + κ, κ ∈ ℂ, that start at initial points z ∈K̂κ(m) ⊂ℂ̂, where ℂ̂ is the set of all rational complex numbers (their real and imaginary parts are rational) and K̂κ(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 K̂κ(m) is a bounded-complexity approximation of the filled Julia set Kκ. 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 fκ is analogous to a universal Turing machine U which executes program p, and an unbounded orbit 𠒪(z,κ) is analogous to an execution of a program p on U that halts. We define a real number Υκ 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 fκ are unbounded. Hence, similar to Chaitin’s Ω number, Υκ acts as a theoretical limit or a “fractal oracle number†that provides an arbitrarily accurate complexity-based approximation of the filled Julia set Kκ. We present a procedure that, when given m and κ, it uses Υκ to generate K̂κ(m). Several numerical examples of sets that estimate K̂κ(m) are presented.