No smooth Julia sets for polynomial diffeomorphisms of $\mathbb{C}^2$ with positive entropy

TL;DR

This paper proves that for polynomial diffeomorphisms of ^2 with positive entropy, neither the Julia set nor its inverse's Julia set can be smooth manifolds, highlighting their complex fractal structure.

Contribution

It establishes the non-smoothness of Julia sets for a broad class of polynomial diffeomorphisms with positive entropy, extending understanding of their geometric complexity.

Findings

Julia sets are not $C^1$ smooth manifolds

Both the Julia set of $f$ and $f^{-1}$ lack smoothness

Results apply to all polynomial diffeomorphisms with positive entropy

Abstract

For any polynomial diffeomorphism $f$ of $\mathbb{C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is $C^1$ smooth as a manifold-with-boundary.