## Abstract

Consider orbits O(z, κ) of the fractal iterator f_{κ}(z):= z^{2} + 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.

Original language | American English |
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Article number | 2450029 |

Journal | Fractals |

Volume | 32 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2024 |

Externally published | Yes |

## Keywords

- Chaitin’s Ω
- Complex Dynamics
- Computation Theory
- Fractal Sets

## All Science Journal Classification (ASJC) codes

- Modelling and Simulation
- Geometry and Topology
- Applied Mathematics