Multiplier rigidity for complex H\'enon maps

TL;DR

This paper proves that complex Hénon maps are uniquely determined by their multiplier spectra, extending classical one-dimensional results to higher dimensions, with implications for understanding stability and parameter space structure.

Contribution

It establishes multiplier rigidity for complex Hénon maps and their compositions, showing they are determined by multipliers, extending McMullen's classical results to higher dimensions.

Findings

Hénon maps are determined by their multiplier spectrum.

Rigidity holds for compositions with fixed multi-degree and multi-Jacobian.

Asymptotic bounds for Lyapunov exponents are crucial for the proof.

Abstract

We investigate the multiplier rigidity problem for polynomial automorphisms of $\mathbf{C}^2$. A first result states that a complex H\'enon 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 H\'enon 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.