For a class of quadratic polynomial endomorphisms f : ℂ2 → ℂ2 close to the standard torus map (x, y) → (x2, y2), 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).
|Original language||English (US)|
|Number of pages||21|
|State||Published - 1998|
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory