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.

考虑轨道${𝒪}（z,\kappa ）$分形迭代器$F_{\kappa }（z）:=z^2+\kappa$,$\kappa \epsilon ℂ$，从初始点开始$z\epsilon {\widehat{K}}_{\kappa }^{（米）}\subset \widehat{ℂ}$， 在哪里$\widehat{ℂ}$是所有有理复数的集合（它们的实部和虚部都是有理数）并且${\widehat{K}}_{\kappa }^{（米）}$由所有此类组成$z$其复杂度不超过某个复杂度参数值$米$（复杂度为$z$被定义为足以描述实部和虚部的位数$z$最低形式）。套装${\widehat{K}}_{\kappa }^{（米）}$是填充 Julia 集的有界复杂度近似值$K_{\kappa }$。我们基于与 Chaitin 的算法信息理论的类比，提出了分形的新视角，其中有理复数$z$是一个程序的模拟$p$, 一个迭代器$F_{\kappa }$类似于通用图灵机$U$执行程序$p$，以及无界轨道${𝒪}（z,\kappa ）$类似于程序的执行$p$在$U$那停止了。我们定义一个实数${\mathrm{{\rm Y}}}_{\kappa }$类似于 Chaitin 的$\mathrm{\Omega }$数字、位置，而不是基于所有程序$p$其执行于$U$停止，它基于所有有理复数$z$其轨道在$F_{\kappa }$是无界的。因此，类似于 Chaitin 的$\mathrm{\Omega }$数字，${\mathrm{{\rm Y}}}_{\kappa }$充当理论极限或“分形预言数”，为填充的 Julia 集提供任意准确的基于复杂性的近似值$K_{\kappa }$。我们提出一个程序，当给出$米$和$\kappa$， 它用${\mathrm{{\rm Y}}}_{\kappa }$生成${\widehat{K}}_{\kappa }^{（米）}$。估计集合的几个数值例子${\widehat{K}}_{\kappa }^{（米）}$被提出。