FRACTAL ORACLE NUMBERS

Joel Ratsaby ()
Additional contact information
Joel Ratsaby: Ariel University, Ariel, Israel

FRACTALS (fractals), 2024, vol. 32, issue 01, 1-15

Abstract: 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.

Keywords: Fractal Sets; Chaitin’s Ω; Computation Theory; Complex Dynamics (search for similar items in EconPapers)
Date: 2024
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.worldscientific.com/doi/abs/10.1142/S0218348X24500294
Access to full text is restricted to subscribers

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wsi:fracta:v:32:y:2024:i:01:n:s0218348x24500294

Ordering information: This journal article can be ordered from

DOI: 10.1142/S0218348X24500294

Access Statistics for this article

FRACTALS (fractals) is currently edited by Tara Taylor

More articles in FRACTALS (fractals) from World Scientific Publishing Co. Pte. Ltd.
Bibliographic data for series maintained by Tai Tone Lim ().