# Real quadratic Julia sets can have arbitrarily high complexity

## TL;DR

This paper demonstrates that certain real quadratic Julia sets can have arbitrarily high computational complexity, surpassing any given threshold, which is a novel finding in the field of complex dynamics.

## Contribution

It establishes the existence of real parameters for quadratic maps whose Julia sets are non-poly-time computable, revealing new complexity properties of these fractals.

## Key findings

- Existence of real parameters with arbitrarily high complexity Julia sets
- First known class of non-poly-time computable Julia sets for real parameters
- Complexity of Julia sets can exceed any fixed threshold

## Abstract

We show that there exist real parameters $c$ for which the Julia set $J_c$ of the quadratic map $z^2+c$ has arbitrarily high computational complexity. More precisely, we show that for any given complexity threshold $T(n)$, there exist a real parameter $c$ such that the computational complexity of computing $J_c$ with $n$ bits of precision is higher than $T(n)$. This is the first known class of real parameters with a non poly-time computable Julia set.