A rigidity theorem for Hénon maps

A rigidity theorem for Hénon maps

“…where C(x, y) = (δ + x, δ − y) with |δ ± | = 1. Though in [5] we did not explicit mention about the presence of the constants c j 's in H, but there presence do not significantly change any computation (see Lemma 3.1). The proof of this result crucially relied on techniques developed by Buzzard and Fornaess in [6], the rigidity theorem of Dinh and Sibony in [7], and introducing the notion of Böttcher coordinates for Hénon maps of the form (1.1), inspired from the construction of Hubbard and Oberste-Vorth in [10].…”

“…We continue to explore the rigidity properties of Hénon maps from [5]. The main motivation for [5] was a result from the dynamics of polynomial maps in one variable by Beardon [1], namely if two polynomials P and Q of degree greater than or equal to 2, have the same Julia set, i.e., J P = J Q then P • Q = σ • Q • P where σ(z) = az + b with |a| = 1 and σ(J P ) = J P . In [5], we provide an analogue of this result for Hénon maps in C 2 .…”

“…The main motivation for [5] was a result from the dynamics of polynomial maps in one variable by Beardon [1], namely if two polynomials P and Q of degree greater than or equal to 2, have the same Julia set, i.e., J P = J Q then P • Q = σ • Q • P where σ(z) = az + b with |a| = 1 and σ(J P ) = J P . In [5], we provide an analogue of this result for Hénon maps in C 2 . To explain it in detail, let us revisit the notations first.…”

“…H ±n (z) is bounded for every n ≥ 1}. Theorem 1.1 of [5] says that, if an automorphism F preserves the non-escaping sets of a Hénon map, i.e., F (K ± H ) = K ± H then F or F −1 is a Hénon map of the form (1.1) and further, they commute upto a linear map. In particular,…”

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The purpose of this note is to explore further the rigidity properties of Hénon maps from [5]. For instance, we show that if H and F are Hénon maps with the same Green measure (µH = µF ), or the same filled Julia set (KH = KF ), or the same Green function (GH = GF ), then H 2 and F 2 have to commute.

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