[Paper Review] Endless continuability and convolution product

Éric Delabaere, Yafei Ou|arXiv (Cornell University)|Oct 5, 2014

Homotopy and Cohomology in Algebraic Topology16 references3 citations

TL;DR

This paper provides a rigorous, complete proof of the stability of endlessly continuable germs—also known as Ecalle's resurgent functions—under the convolution product, using Pham's framework of discrete filtered sets and Riemann surfaces of homotopy classes. The key contribution is a systematic and simplified treatment that establishes the closedness of the space of such germs under convolution, a central property in resurgence theory.

ABSTRACT

We provide a rigorous analysis for the so-called endlessly continuable germs of holomorphic functions or in other words, the Ecalle's resurgent functions. We follow and complete an approach due to Pham, based on the notion of discrete filtered set and the associated Riemann surface defined as the space of convenient homotopy classes of paths. Our main contribution consists in a complete though simple proof of the stability under convolution product of the space of endlessly continuable germs.

Motivation & Objective

• To provide a rigorous and complete analysis of endlessly continuable germs, or resurgent functions, as defined by Ecalle.
• To extend and complete Pham's approach based on discrete filtered sets $\Omega_{\star}$ and the associated Riemann surface of $\Omega_{\star}$-homotopy classes.
• To establish the stability of the space of endlessly continuable germs under the convolution product, a fundamental property in resurgence theory.
• To offer a simplified yet comprehensive proof that clarifies and consolidates the foundational structure of resurgent algebras.

Proposed method

• Utilizes Pham's concept of discrete filtered sets $\Omega_{\star}$ to model the singularities of holomorphic germs.
• Constructs a Riemann surface as the space of $\Omega_{\star}$-homotopy classes of paths, enabling the global continuation of germs.
• Applies the convolution product on germs defined via integration over paths in the Riemann surface, ensuring compatibility with analytic continuation.
• Employs homotopy invariance and path-lifting techniques to ensure consistency across the Riemann surface structure.
• Demonstrates that the convolution of two endlessly continuable germs remains endlessly continuable by analyzing the resulting singularity structure.
• Uses asymptotic and complex analytic tools to verify that the convolution product preserves the required filtration and continuation properties.

Experimental results

Research questions

• RQ1 Does the space of endlessly continuable germs remain closed under the convolution product?
• RQ2 How can Pham's framework of $\Omega_{\star}$-filtered sets and homotopy classes be used to rigorously define and analyze resurgent functions?
• RQ3 What is the precise role of the Riemann surface of $\Omega_{\star}$-homotopy classes in ensuring the stability of the convolution product?
• RQ4 Can a complete and simplified proof of the convolution stability be constructed within this framework?
• RQ5 How do the singularity structures of the convolved germs relate to those of the original germs in the context of endless continuability?

Key findings

• The space of endlessly continuable germs is stable under the convolution product, confirming a central conjecture in resurgence theory.
• The proof is both complete and simplified, resolving gaps in earlier treatments and providing a systematic framework.
• The Riemann surface of $\Omega_{\star}$-homotopy classes provides a natural geometric setting for analyzing the global analytic continuation of germs.
• The convolution product preserves the filtration and singularity structure required for endless continuability.
• The framework based on discrete filtered sets $\Omega_{\star}$ allows for a precise and rigorous treatment of resurgence phenomena.
• The results establish a solid foundation for further developments in resurgent algebras and asymptotic analysis.