[Paper Review] Normalization of bundle holomorphic contractions and applications to dynamics

François Berteloot, Christophe Dupont+1 more|ArXiv.org|May 18, 2007

Mathematical Dynamics and Fractals3 citations

TL;DR

This paper establishes a non-autonomous Poincaré-Dulac theorem for bundle maps of holomorphic contractions with regular splitting of contraction rates, proving convergence of iterated conjugation maps under resonant conditions on the exponential contraction moduli. The key result enables precise distortion estimates of ellipsoids along typical orbits in complex dynamics, leading to approximation of Lyapunov exponents via multipliers of repelling cycles.

ABSTRACT

We establish a Poincaré-Dulac theorem for sequences (G_n)_n of holomorphic contractions whose differentials d_0 G_n split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps. Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of CP(k). In this context, our normalization result allows to precisely estimate the distortions of ellipsoids along typical orbits. As an application, we show how the Lyapunov exponents of the equilibrium measure are approximated in terms of the multipliers of the repulsive cycles.

Motivation & Objective

• To extend the classical Poincaré-Dulac normalization theorem to non-autonomous systems of holomorphic contractions in a bundle map framework.
• To characterize the resonant relations governing normal forms in terms of the moduli of exponential contraction rates rather than eigenvalues.
• To provide precise distortion estimates for ellipsoids under iterated inverse branches of endomorphisms on complex projective space.
• To establish a connection between the Lyapunov exponents of the equilibrium measure and the multipliers of repelling cycles via the normalization process.

Proposed method

• Formalizing the setting using bundle maps over a bijective base transformation τ, with fibers modeled on C^k and tubes defined by ε-slow functions.
• Introducing the notion of tame bundle maps, where Taylor coefficients grow slowly enough to allow convergence of iterated conjugation sequences.
• Defining regular contracting linear bundle maps via decomposition into invariant subbundles with distinct exponential contraction rates Λ_j and uniform error bounds.
• Establishing convergence of the sequence T_n = N^{-n} F^n to a limit map T that is κ-tangent to the identity at order q+1, under tangency and resonance conditions.
• Using spectral estimates on exterior powers of the derivative to control the growth of volume distortion and derive Lyapunov exponent approximations.
• Applying Hadamard’s inequality and block-triangularization techniques to bound the norms of wedge powers of the derivative along orbits.

Experimental results

Research questions

• RQ1 Under what conditions can a sequence of holomorphic contractions on a bundle be conjugated to a normal form defined by resonant monomials?
• RQ2 How do resonant relations among contraction rates Λ_j affect the convergence of the normalization process in the non-autonomous setting?
• RQ3 Can the Lyapunov exponents of the equilibrium measure in complex dynamics be approximated using only the multipliers of repelling cycles?
• RQ4 What role do ε-slow functions play in ensuring uniform contraction and convergence of the conjugation map across the base space?

Key findings

• The normalization map T = lim_{n→∞} N^{-n} F^n converges uniformly on ε-slow tubes E(ρ_ε), with T being κ-tangent to the identity at order q+1.
• For any θ ∈ (M^{q+1}/m, 1), the error in the conjugation satisfies |(N^n ∘ T - F^n)(v)| ≤ φ_ε(x)(mθ)^n |v|^{q+1} for some ε-fast function φ_ε.
• The Lyapunov exponents of the equilibrium measure are approximated by the logarithms of the multipliers of repelling cycles, with error bounded by O(1/n) + ηε.
• The norm of the s-th exterior power of the derivative |∧^s d_w R^n_x| satisfies |(1/n) log |∧^s d_w R^n_x| - (λ_1 + … + λ_s)| ≤ (1/n) log H_{ηε}(x) + ηε for η = kθ and H_{ηε} = (M'_{θε})^k.
• The convergence and distortion estimates hold under the condition M^{q+1} < m, where m and M are the infimum and supremum of the contraction rates of the linear part.
• The block-diagonalization of the linear part A_x^n via unitary transformations preserves the essential spectral estimates, enabling control over volume growth in each invariant subspace.