[Paper Review] Equidistribution of saddle periodic points for Henon-type automorphisms of C^k

Tien‐Cuong Dinh, Nessim Sibony|arXiv (Cornell University)|Mar 1, 2014

Mathematical Dynamics and Fractals14 references5 citations

TL;DR

This paper establishes the equidistribution of saddle periodic points with respect to the equilibrium measure for Hénon-type automorphisms on ℂ^k, using a novel approach based on the theory of densities for positive closed currents. The key result shows that the normalized counting measures of saddle periodic points of period n converge weakly to the Green (equilibrium) measure μ as n → ∞.

ABSTRACT

In this paper, we prove the equidistribution of saddle periodic points for Henon-type automorphisms of C^k with respect to it equilibrium measure. A general strategy to obtain equidistribution properties in any dimension is presented. It is based on our recent theory of densities for positive closed currents. Several fine properties of dynamical currents are also proved.

Motivation & Objective

• To establish the equidistribution of saddle periodic points with respect to the equilibrium measure μ for Hénon-type automorphisms on ℂ^k.
• To develop a general strategy for proving equidistribution in arbitrary dimension using advanced tools from complex dynamics and current theory.
• To extend the equidistribution result beyond the classical case of dimension 2 to higher-dimensional holomorphic automorphisms with non-uniform hyperbolicity.
• To prove fine properties of dynamical currents, particularly the existence and structure of tangent currents and their intersections.
• To provide a framework applicable to other meromorphic dynamical systems beyond Hénon maps.

Proposed method

• Utilizes the theory of densities for positive closed currents, particularly the notion of tangent currents and weak intersections.
• Applies the convergence of pullbacks of the Fubini-Study form to define Green currents τ₊ and τ₋, which generate the main dynamical currents T₊ = τ₊^p and T₋ = τ₋^{k−p}.
• Constructs the equilibrium measure μ as the wedge product μ = T₊ ∧ T₋ = τ₊^p ∧ τ₋^{k−p}, which is invariant and of maximal entropy log d.
• Analyzes the intersection of the graph of fⁿ with the diagonal Δ in ℙ^k × ℙ^k to identify periodic points as solutions to fⁿ(z) = z.
• Uses the current 𝕊 = lim d⁻ⁿ ∑[Γₙ^{(j)}] as a limit of normalized graphs to construct a comparison measure μ′′Δ ≤ μΔ and μ′′Δ ≤ μ′Δ.
• Applies Proposition 5.13 to bound the mass of the constructed measure μ′′Δ from below by 1−δ, ensuring convergence to μ.

Experimental results

Research questions

• RQ1 Do saddle periodic points of Hénon-type automorphisms on ℂ^k equidistribute with respect to the equilibrium measure μ?
• RQ2 Can the equidistribution of periodic points be established in arbitrary dimension k ≥ 2 using current-theoretic methods?
• RQ3 What is the role of the Green currents T₊ and T₋ in the equidistribution of periodic points?
• RQ4 How do the indeterminacy sets I₊ and I₋ affect the dynamics and the construction of invariant measures?
• RQ5 Can the method be generalized to other intersections, such as f⁻ⁿ(L₊) ∩ fⁿ(L₋), for analytic subvarieties L₊, L₋?

Key findings

• The normalized counting measure d⁻ⁿ ∑_{a∈SPₙ^ε} δₐ converges weakly to the equilibrium measure μ as n → ∞.
• The equilibrium measure μ = T₊ ∧ T₋ is the unique invariant probability measure of maximal entropy log d for f and f⁻¹.
• The mass of the comparison measure μ′′Δ constructed via tangent currents is bounded below by 1−δ for any δ > 0, ensuring convergence to μ.
• The limit current 𝕊 is supported on lames that are locally products of stable and unstable manifolds, reflecting the hyperbolic structure of μ.
• The method applies to more general intersections: points in f⁻ⁿ(L₊) ∩ fⁿ(L₋) are equidistributed with respect to μ under suitable transversality conditions.
• The theory of densities and weak intersections of currents enables equidistribution results in non-uniformly hyperbolic settings with arbitrary numbers of stable and unstable directions.