Multiplier rigidity for complex Hénon maps

We investigate the multiplier rigidity problem for polynomial automorphisms of $\mathbf{C}^2$. A first result states that a complex Hénon map of given degree is determined up to finitely many choices by its multiplier spectrum, or more generally by the unstable multipliers of its saddle periodic points. This is the counterpart in this setting of a classical result of McMullen for one-dimensional rational maps. For compositions of Hénon maps, the same rigidity holds provided the multi-degree and the multi-Jacobian are fixed. As in McMullen’s theorem, this follows from the nonexistence of stable algebraic families in the corresponding parameter space. This in turn relies on precise asymptotic bounds for the Lyapunov exponents of the maximal entropy measure along diverging families.

💡 Research Summary

The paper studies the “multiplier rigidity” problem for polynomial automorphisms of the complex plane ℂ², focusing on complex Hénon maps and their compositions. A multiplier is an eigenvalue of the derivative of an iterate at a periodic point; the authors consider two spectral data sets: the full trace spectrum (trace = λ₁ + λ₂ of Dfⁿ at each periodic point) and the unstable multiplier spectrum (the eigenvalue λ_u with |λ_u| > 1 for saddle cycles).

The authors first recall the classification of polynomial automorphisms: those with dynamical degree λ₁ = 1 are elementary, while λ₁ > 1 are called loxodromic. Every loxodromic map can be conjugated to a composition of finitely many monic, centered Hénon maps h_i(x,y) = (a_i y + p_i(x), x). The ordered list of degrees (d₁,…,d_k) is the multidegree, and the list of constants (a₁,…,a_k) is the multi‑Jacobian. Up to permutation of the factors and a finite diagonal action of the group of (d_i – 1)‑th roots of unity, this normal form is unique.

Theorem A (single Hénon map) states that a Hénon map of fixed degree d is determined up to finitely many possibilities by its trace spectrum, and the same holds for the unstable multiplier spectrum. In particular, for a generic Hénon map the spectrum determines the map uniquely (up to conjugacy). Remarkably, the Jacobian does not need to be known.

Theorem B extends the result to compositions of Hénon maps. If the multidegree and the multi‑Jacobian are fixed, then the trace (or unstable multiplier) spectrum again determines the map up to finitely many choices.

Theorem C addresses C¹‑rigidity: if two maps of the same (multi)degree are C¹‑conjugate on a neighbourhood of their Julia set and are sufficiently close, then they are actually equal. This gives a local rigidity statement and a global finiteness result for the C¹‑conjugacy class.

The backbone of these rigidity statements is Theorem D, which asserts that any stable irreducible algebraic family of Hénon maps (or of compositions with fixed multi‑Jacobian) must be trivial, i.e., its image reduces to a single point. “Stable” here means weak stability in the sense of Lyubich–Dujardin: the multipliers of all saddle periodic points vary holomorphically and stay away from the unit circle. The proof follows McMullen’s strategy for one‑dimensional rational maps: a non‑trivial stable family would have to be unbounded in parameter space, leading to a degeneration phenomenon.

The key quantitative tool for ruling out such degeneration is Theorem E. For a composition f = h_k ∘ … ∘ h₁ with multidegree d = (d₁,…,d_k) and multi‑Jacobian a = (a₁,…,a_k), define the “maximal escape rate”

M(f) = max { M(p_i) | i = 1,…,k },

where M(p) = max { G_p(c) | c critical of p } is the usual one‑dimensional escape rate of the polynomial p_i. Theorem E proves the existence of constants C₁, C₂ > 0 (depending only on d and a) such that for every f with M(f) ≥ 1

C₁ M(f) ≤ χ⁺(μ_f) ≤ C₂ M(f),

where χ⁺(μ_f) is the larger Lyapunov exponent of the canonical equilibrium measure μ_f of f. An analogous bound holds for the smaller exponent χ⁻(μ_f) because χ⁺ + χ⁻ = log|Jac f|. This estimate is a two‑dimensional analogue of the Manning‑Przytycki formula for one‑dimensional polynomials, and its proof combines Bedford–Smillie’s exact Lyapunov formula with Huguin’s one‑dimensional estimates and a careful analysis of “unstable critical points”.

With Theorem E, a stable algebraic family would force M(f) to stay bounded; however M is a proper map on the parameter space, so boundedness forces the family to be finite, and irreducibility then implies triviality. Consequently, the parameter sets defined by fixing the trace (or unstable multiplier) spectra are stable algebraic families, and Theorem D forces them to be singletons (or at most a finite set). This yields Theorems A, B, and C.

An additional corollary is the compactness of the connectedness locus (the set of parameters for which the Julia set is connected) when the multi‑Jacobian is fixed. Theorem E together with results of Dujardin–Lyubich and Bedford–Smillie shows that M(f) = 0 exactly when the Julia set is connected, and because M is proper, the locus is compact. The paper also exhibits examples where varying the Jacobian destroys this compactness, highlighting a genuine difference between one‑ and two‑dimensional dynamics.

Overall, the work extends McMullen’s multiplier rigidity from one‑dimensional rational maps to the richer setting of complex Hénon dynamics. It establishes that the spectral data of periodic points (traces or unstable multipliers) essentially determines the map, even without knowledge of the Jacobian, and it provides new quantitative relations between Lyapunov exponents and escape rates in two dimensions. The techniques blend algebraic geometry (stability of families), pluripotential theory (Green functions, equilibrium measures), and fine dynamical estimates, opening avenues for further study of non‑Archimedean degenerations and higher‑dimensional rigidity phenomena.