Fractal oracle numbers

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

Original languageEnglish
Article number2450029
Issue number1
StatePublished - 2024


  • Chaitin's ω
  • Complex Dynamics
  • Computation Theory
  • Fractal Sets


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