Julia sets for Fibonacci endomorphisms of \(\mathbb R^2\). (English) Zbl 1402.37058
Summary: We study the dynamics of the family \(f_c(x,y)=(xy+c,x)\) of endomorphisms of \(\mathbb R^2\), 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 \(0 < c <\frac{1}{4}\), these two filled Julia sets can be described explicitly as finite unions of invariant manifolds.
MSC:
| 37F45 | Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) |
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