[Paper Review] Invariants of identity-tangent diffeomorphisms: explicit formulae and effective computation

Olivier Bouillot, Jean Écalle|arXiv (Cornell University)|Apr 3, 2014

Mathematical Dynamics and Fractals3 citations

TL;DR

This paper presents explicit formulae and effective computational methods for scalar invariants of identity-tangent diffeomorphisms on ℂ, using multitangents, multizetas, and resurgent functions. It establishes a systematic framework to compute analytic invariants via connectors and collectors, with closed-form expansions up to weight 15, revealing deep connections between zeta values and dynamical invariants.

ABSTRACT

In this short Survey we revisit the subject of local, identity-tangent diffeomorphisms of $\doC$ and their analytic invariants, under two viewpoints: that of explicit expansions, which necessarily involve multitangents and multizetas; and that of effective computation. Along the way, we stress the difference between the extup{collectors} (pre-invariant but of one piece) and the extup{connectors} (invariant but mutually unrelated). We also attempt to streamline the nomenclature and notations.

Motivation & Objective

• To develop explicit, computable expressions for scalar analytic invariants of identity-tangent diffeomorphisms in complex dynamics.
• To clarify the distinction between collectors (pre-invariants) and connectors (true invariants) in the context of resurgent analysis.
• To streamline notation and nomenclature for multitangents, multizetas, and resurgent monomials in the study of diffeomorphisms.
• To provide algorithmic and symbolic methods for computing invariants up to high weight, including explicit expansions in zeta and multiple zeta values.
• To extend the theory to ramified and reflexive cases, with normalization and parity considerations.

Proposed method

• The paper employs mould calculus and symmetries (e.g., symmetrel, alternal) to express multitangents in terms of multizeta values and resurgent monomials.
• It introduces a direct, non-symmetrical scheme and an indirect, symmetrical scheme for computing collectors and connectors from the diffeomorphism f.
• The infinitesimal generators f∗ and π∗ are used to define the connectors via exponential maps involving divergent formal series.
• Explicit expansions of invariants Aω are derived as entire functions of f, using recursive relations and normalization for ρ(f) ≠ 0 or p(f) ≠ 1.
• The method incorporates parity separation and handles divergent and ramified cases through normalization and rescaling techniques.
• Tables of multitangents and invariants are constructed to systematize the computation of invariants up to weight 15.

Experimental results

Research questions

• RQ1 How can scalar invariants of identity-tangent diffeomorphisms be expressed explicitly in terms of the diffeomorphism f?
• RQ2 What is the precise relationship between multitangents, multizetas, and the invariants Aω?
• RQ3 How do collectors and connectors differ in their transformation properties and roles in the invariant structure?
• RQ4 What computational strategies allow for effective, high-weight expansion of invariants involving zeta and multiple zeta values?
• RQ5 How do normalization and ramification affect the structure of invariants in the general case?

Key findings

• The paper provides explicit expansions of invariants Aω as entire functions of f, with coefficients expressed in terms of multizeta values and multitangents up to weight 15.
• For f = l ∘ g with g(z) = z(1 + g₂z⁻²)¹ᐟ², the invariant 𝔉∗ is computed up to weight 12, with terms involving ζ(3), ζ(5), ζ(7), and ζ(9).
• In the case g(z) = z(1 + 3g₃z⁻³)¹ᐟ³, the invariant 𝔉∗ is computed up to weight 15, including complex combinations of ζ(2)⁶, ζ(3)⁴, ζ(5)², and ζ(10,2).
• The method successfully computes invariants with coefficients involving multiple zeta values such as ζ(6,2), ζ(8,2), and ζ(10,2), demonstrating the role of algebraic relations in simplifying expressions.
• The framework allows for the systematic computation of invariants in reflexive and unitary cases, with parity-separated expansions.
• The results confirm that invariants are entire functions of f, with controlled growth properties and explicit dependence on zeta values and their products.