FRACTAL ORACLE NUMBERS

Consider orbits ${𝒪}(z,\kappa )$ of the fractal iterator $f_{\kappa }(z):=z^2+\kappa$, $\kappa \in ℂ$, that start at initial points $z\in {\widehat{K}}_{\kappa }^{(m)}\subset \widehat{ℂ}$, where $\widehat{ℂ}$ is the set of all rational complex numbers (their real and imaginary parts are rational) and ${\widehat{K}}_{\kappa }^{(m)}$ 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 ${\widehat{K}}_{\kappa }^{(m)}$ is a bounded-complexity approximation of the filled Julia set $K_{\kappa }$. 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 $f_{\kappa }$ is analogous to a universal Turing machine $U$ which executes program $p$, and an unbounded orbit ${𝒪}(z,\kappa )$ is analogous to an execution of a program $p$ on $U$ that halts. We define a real number ${\mathrm{{\rm Y}}}_{\kappa }$ which resembles Chaitin’s $\mathrm{\Omega }$ 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 $f_{\kappa }$ are unbounded. Hence, similar to Chaitin’s $\mathrm{\Omega }$ number, ${\mathrm{{\rm Y}}}_{\kappa }$ acts as a theoretical limit or a “fractal oracle number” that provides an arbitrarily accurate complexity-based approximation of the filled Julia set $K_{\kappa }$. We present a procedure that, when given $m$ and $\kappa$, it uses ${\mathrm{{\rm Y}}}_{\kappa }$ to generate ${\widehat{K}}_{\kappa }^{(m)}$. Several numerical examples of sets that estimate ${\widehat{K}}_{\kappa }^{(m)}$ are presented.