Jordan tori and polynomial endomorphisms in $ℂ^2$
Volume 157 / 1998
Fundamenta Mathematicae 157 (1998), 139-159 DOI: 10.4064/fm-157-2-3-139-159
For a class of quadratic polynomial endomorphisms $f: ℂ^2 → ℂ^2$ close to the standard torus map $(x,y) → (x^2,y^2)$, we show that the Julia set J(f) is homeomorphic to the torus. We identify J(f) as the closure ℛ of the set of repelling periodic points and as the Shilov boundary of the set K(f) of points with bounded forward orbit. Moreover, it turns out that (J(f),f) is a mixing repeller and supports a measure of maximal entropy for f which is uniquely determined as the harmonic measure for K(f).