Mathematical Research Letters

Volume 27 (2020)

Number 3

Saddle hyperbolicity implies hyperbolicity for polynomial automorphisms of $\mathbb{C}^2$

Pages: 693 – 709



Romain Dujardin (Laboratoire de probabilités, statistique et modélisation, Sorbonne Universités, Paris, France)


We prove that for a polynomial diffeomorphism of $\mathbb{C}^2$, uniform hyperbolicity on the set of saddle periodic points implies that saddle points are dense in the Julia set. In particular $f$ satisfies Smale’s Axiom A on $\mathbb{C}^2$.

This research was partially supported by ANR project LAMBDA, ANR-13-BS01-0002 and a grant from the Institut Universitaire de France.

Received 26 September 2018

Accepted 3 April 2019

Published 20 August 2020