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Resurgent functions and splitting problems

David Sauzin

TL;DR

This paper surveys Écalle's theory of resurgent functions and alien calculus, presenting a self-contained treatment of the algebraic and analytic framework that connects divergent series to analytic continuations via the formal Borel transform and sectorial sums. It develops Abel's equation as a leading nonlinear example, explaining how Fatou coordinates and horn maps encode a nonlinear Stokes phenomenon and how alien derivations produce bridge equations linking singularities to classical derivatives. The survey then formalizes singularities through the SING/ANA framework and extends these to general resurgent functions with alien derivations, culminating in an algebraic portrait of the resurgent world and its invariants. Finally, it applies these tools to splitting problems in second-order and real two-dimensional dynamics, including parametric resurgence phenomena and the inner–outer decomposition that governs exponentially small separations of invariant foliations, with explicit invariants $A_{\omega}$ surfacing as key data.

Abstract

The present text is an introduction to Écalle's theory of resurgent functions and alien calculus, in connection with problems of exponentially small separatrix splitting. An outline of the resurgent treatment of Abel's equation for resonant dynamics in one complex variable is included. The emphasis is on examples of nonlinear difference equations, as a simple and natural way of introducing the concepts.

Resurgent functions and splitting problems

TL;DR

This paper surveys Écalle's theory of resurgent functions and alien calculus, presenting a self-contained treatment of the algebraic and analytic framework that connects divergent series to analytic continuations via the formal Borel transform and sectorial sums. It develops Abel's equation as a leading nonlinear example, explaining how Fatou coordinates and horn maps encode a nonlinear Stokes phenomenon and how alien derivations produce bridge equations linking singularities to classical derivatives. The survey then formalizes singularities through the SING/ANA framework and extends these to general resurgent functions with alien derivations, culminating in an algebraic portrait of the resurgent world and its invariants. Finally, it applies these tools to splitting problems in second-order and real two-dimensional dynamics, including parametric resurgence phenomena and the inner–outer decomposition that governs exponentially small separations of invariant foliations, with explicit invariants surfacing as key data.

Abstract

The present text is an introduction to Écalle's theory of resurgent functions and alien calculus, in connection with problems of exponentially small separatrix splitting. An outline of the resurgent treatment of Abel's equation for resonant dynamics in one complex variable is included. The emphasis is on examples of nonlinear difference equations, as a simple and natural way of introducing the concepts.

Paper Structure

This paper contains 18 sections, 28 theorems, 249 equations, 13 figures.

Table of Contents

  1. The algebra of resurgent functions
  2. Formal Borel transform
  3. Linear and nonlinear difference equations
  4. The Riemann surface $\mathcal{R}$ and the analytic continuation of convolution
  5. Formal and convolutive models of the algebra of resurgent functions, $\tilde{\mathcal{H}}$ and $\widehat{\mathcal{H}}(\mathcal{R})$
  6. Alien calculus and Abel's equation
  7. Abel's equation and tangent-to-identity holomorphic germs of $(\mathbb{C},0)$
  8. Sectorial normalisations (Fatou coordinates) and nonlinear Stokes phenomenon (horn maps)
  9. Alien calculus for simple resurgent fuctions
  10. Bridge equation for non-degenerate parabolic germs
  11. Formalism of singularities, general resurgent functions and alien derivations
  12. General singularities. Majors and minors. Integrable singularities
  13. The convolution algebra $\operatorname{SING}$
  14. General resurgent functions and alien derivations
  15. Splitting problems
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\hat{\varphi}$ and $\hat{\psi}$ denote the formal Borel transforms of $\tilde{\varphi},\tilde{\psi}\in z^{-1}\mathbb{C}[[z^{-1}]]$ and consider the product series $\tilde{\chi}=\tilde{\varphi}\tilde{\psi}$. Then its formal Borel transform is given by the "convolution"

Figures (13)

  • Figure 1: Laplace integral in the direction ${\theta}$ gives rise to functions analytic in the half-plane $\mathop{\Re e}\nolimits (z\,\mathrm e^{{\mathrm i}{\theta}})>\tau$.
  • Figure 2: Sectorial sums.
  • Figure 3: Several Borel-Laplace sums, analytic in different domains, may be attached to a single divergent series.
  • Figure 4: Borel-Laplace summation for the difference equation (\ref{['eqnph']}).
  • Figure 5:
  • ...and 8 more figures

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Definition 4
  • Proposition 1
  • Proposition 2
  • ...and 29 more