# Non computable Mandelbrot-like set for a one-parameter complex family

## TL;DR

This paper demonstrates that for certain computable complex parameters, the bifurcation locus of a specific one-parameter complex family is not Turing computable, revealing limits of algorithmic predictability in complex dynamics.

## Contribution

It establishes the existence of computable parameters where the bifurcation locus is non-Turing computable, highlighting fundamental limits in computational complex dynamics.

## Key findings

- Existence of non-Turing computable bifurcation loci for certain parameters
- Identification of computable complex numbers with non-computable bifurcation sets
- Advancement in understanding the computational complexity of dynamical systems

## Abstract

We show the existence of computable complex numbers $\lambda$ for which the bifurcation locus of the one parameter complex family $f_{b}(z) = \lambda z + b z^{2} + z^{3}$ is not Turing computable.