Julia sets for Fibonacci endomorphisms of ℝ²

ABSTRACT

We study the dynamics of the family f _c(x, y) = (xy + c, x) of endomorphisms of , where c is a real parameter. We investigate several topological properties of the forward and backward filled Julia sets. Then, through the study of adapted dynamical filtrations of the plane, we prove that for the interval of parameters given by 0 < c <, these two filled Julia sets can be described explicitly as finite unions of invariant manifolds.

KEYWORDS:

• Holomorphic dynamics
• Fibonacci maps
• filled Julia sets

MATHEMATICS SUBJECT CLASSIFICATION CODES:

• 37F45

Acknowledgments

The first author was supported by FAPESP grant 2011/12650-4. The second author was supported by FAPESP grant 2011/16265-8. The third author would like to express thanks to the IME-USP (São Paulo) for the warm hospitality during his visit. He was supported by FAPESP grants 2011/23199-1 and 2013/23643-4 and by CNPq grants 307154/2012-2 and 482519/2012-6. He would also like to express thanks to Pierre Arnoux and Patrícia Cirilo Romano for fruitful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

FAPESP [grant number 2011/12650-4], [grant number 2011/16265-8], [grant number 2011/23199-1], [grant number 2013/23643-4]; CNPq [grant number 307154/2012-2], [grant number 482519/2012-6].