Filled Julia set of some class of Hénon-like maps

ABSTRACT

In this work we consider a class of endomorphisms of ${\mathbb{R}}^2$ defined by $f(x,y)=(xy+c,x)$, where $c\in \mathbb{R}$ is a real number and we prove that when $-1<c<0$, the forward filled Julia set of f is the union of stable manifolds of fixed and $3-$periodic points of f. We also prove that the backward filled Julia set of f is the union of unstable manifolds of the saddle fixed and $3-$periodic points of f.

KEYWORDS:

• Holomorphic dynamics
• stable manifolds
• filled Julia sets

2010:

• 37F45
• 37F50

Acknowledgements

I heartily thank Ali Messaoudi for many helpful comments, discussions and suggestions. I also want to thank the Institut de Mathématiques de Luminy, where I have done part of this work. I would also like to thank the anonymous referees for an especially careful reading of this paper and for the useful comments and remarks.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Supported by FAPESP grants 2015/26161-6 and 2018/13720-5.