Jordan tori and polynomial endomorphisms in C2

For a class of quadratic polynomial endomorphisms f : C-2 --> C-2 close to the standard torus map (x, y) bar right arrow (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).