Julia sets for a class of polynomial maps in R^2 and C^2
Aix-Marseille Université - Site St Charles
3, place Victor Hugo - case 39
13331 MARSEILLE Cedex 03
In this lecture we consider a class of endomorphisms of $\mathbbR^2$ defined by $f(x,y)=(xy+c,x)$, where $c\in\mathbbR$ 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 $f$. Furthermore, we prove that the backward filled Julia set of $f$ is the union of unstable manifolds of the saddle fixed and $3-$periodic points of $f$.
We also study the dynamics of the family $f_c,d(x, y) = (xy+c, x+d)$ of endomorphisms of $\mathbbC^2$, where $c$ and $d$ are complex parameters with $|d|<1$.
- Danilo CAPRIO