[Paper Review] Saddle hyperbolicity implies hyperbolicity for polynomial automorphisms of C^2

Romain Dujardin|arXiv (Cornell University)|Jun 28, 2018

Mathematical Dynamics and Fractals6 references3 citations

TL;DR

This paper proves that for polynomial automorphisms of C^2, uniform hyperbolicity on the set of saddle periodic points (J*) implies that J* is dense in the Julia set J, thus establishing J = J*. The result resolves a long-standing open problem by showing that saddle hyperbolicity implies full hyperbolicity in the sense of Bedford and Smillie, using a combination of ergodic closing lemmas, holomorphic motions, and potential-theoretic techniques in both dissipative and conservative settings.

ABSTRACT

We prove that for a polynomial diffeomorphism of C^2, uniform hyperbolicity on the set of saddle periodic points implies that saddle points are dense in the Julia set. In particular f satisfies Smale's Axiom A on C^2 .

Motivation & Objective

• To resolve the open question of whether uniform hyperbolicity on the set of saddle periodic points (J*) implies hyperbolicity of the full Julia set J in polynomial automorphisms of C^2.
• To establish that J = J* under saddle hyperbolicity, thereby confirming that hyperbolicity on J* implies the full Axiom A and laminar structure properties of Bedford and Smillie.
• To extend previous partial results—particularly those under the substantially dissipative condition—to the general case, including conservative maps.
• To provide a unified proof strategy combining ergodic theory, holomorphic motions, and potential theory for both dissipative and conservative settings.

Proposed method

• Uses proof by contradiction: assumes f is dissipative, uniformly hyperbolic on J*, and J* ≠ J.
• Applies the ergodic closing lemma from prior work to show that for some p ∈ J*, the stable manifold W^s(p) intersects J− in a relatively open, non-trivial subset—a phenomenon unexpected in dissipative dynamics.
• Employs results from [BS6] on stable slices of J− and potential-theoretic tools to derive a contradiction in the dissipative case.
• For the conservative case, constructs a holomorphic family (f_λ) with f₀ = f, such that f_λ is dissipative for parameters near 0.
• Uses holomorphic motion extension theorems from [DL] to extend the motion of J* to the entire K-set, preserving disjointness from J*.
• Derives a contradiction by showing that a dissipative parameter f_λ₁ must have an attracting periodic point, contradicting the conservative nature of f₀.

Experimental results

Research questions

• RQ1 Does uniform hyperbolicity on the set of saddle periodic points (J*) imply that the full Julia set J is hyperbolic in polynomial automorphisms of C^2?
• RQ2 Can the result that J = J* be established without assuming substantial dissipation or other restrictive conditions?
• RQ3 What is the role of holomorphic motions and parameter families in extending results from the dissipative to the conservative case?
• RQ4 How do stable manifolds and potential-theoretic structures interact in the presence of hyperbolicity on J*?
• RQ5 Is it possible to derive global laminar structure on J± from hyperbolicity on the countable, dense set J*?

Key findings

• The main result establishes that if f is a polynomial automorphism of C^2 with non-trivial dynamics and uniformly hyperbolic on J*, then J = J*.
• In the dissipative case, the proof shows that J− ∩ W^s(p) contains a relatively open subset for some p ∈ J*, contradicting known results in the substantially dissipative regime.
• For conservative maps, the contradiction arises via a holomorphic motion argument: the existence of a point in K(f₀) \ J*(f₀) extends to a point in K(f_λ) \ J*(f_λ) for nearby dissipative f_λ, implying an attracting periodic point, which contradicts the conservative nature of f₀.
• The result confirms that saddle hyperbolicity implies full hyperbolicity in the sense of Bedford and Smillie, meaning f satisfies Axiom A and has a laminar structure on J±.
• The proof resolves a long-standing open problem previously only partially addressed under the substantially dissipative condition (Jac(f) < d⁻²).
• The work establishes that hyperbolicity on the countable, dynamically defined set J* is sufficient to imply global hyperbolicity and the full dynamical structure of the Julia set.