Symmetries of Julia sets of nondegenerate polynomial skew products on C2

Böttcher coordinates for polynomial skew products

“…Let f (z, w) = (z δ , q(z, w)) be semiconjugate to g(z, w) = (z δ , h(w)) by π(z, w) = (z r , z s w) for some non-zero integers r and s: π [11] and [15], such maps can be characterized by the symmetries of Julia sets.…”

“…For polynomial skew products, we have been using fiberwise Böttcher coordinates; see e.g. [9,6,11] and [15].…”

Polynomial skew products whose Julia sets have infinitely many symmetries

“…Under some assumptions, we characterize Γ f by functional equations including the iterates of f . Although the group Σ p of a polynomial p is characterized by the unique equation pσ = σ δ p, our characterization of Γ f needs infinitely many equations as in [3,Lemma 3.2]. Referring to Proposition 4.2, we define d).…”

“…The study of the symmetries of Julia sets begins in Section 4. First, we define the centroids of f as defined in [1] and [3], and show that the symmetries of the Julia set of f are birationally conjugate to rotational products. The tools for the proof are the fiberwise Green and Böttcher functions of f , which also relate to the centroids of f .…”

“…Beardon [1] investigated the symmetries of the Julia sets of polynomials on C. He considered conformal functions as symmetries, and proved that the group of symmetries is infinite if and only if the polynomial is conjugate to z → z δ . To generalize the results in one-dimension to those in higher dimensions, we [3] previously investigated the symmetries of Julia sets of nondegenerate polynomial skew products on C 2 . We defined the Julia sets as the supports of the Green measures, which are compact, and considered suitable polynomial automorphisms as symmetries.…”