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On the monograph "Finiteness Theorems for limit cycles" and a special case of alternant cycles

Melvin Yeung

Abstract

We provide evidence that the approach of [Ilyashenko 1991] to the proof of Dulac's theorem has a gap. Although the asymptotics of [Ilyashenko 1991] capture far more than the asymptotics of Dulac, we prove that the arguments for why the asymptotics in [Ilyashenko 1991] are not themselves oscillatory is insufficient. We give an explicit counterexample and we draw confines to which Ilyashenko's result may be restricted in order to keep the validity.

On the monograph "Finiteness Theorems for limit cycles" and a special case of alternant cycles

Abstract

We provide evidence that the approach of [Ilyashenko 1991] to the proof of Dulac's theorem has a gap. Although the asymptotics of [Ilyashenko 1991] capture far more than the asymptotics of Dulac, we prove that the arguments for why the asymptotics in [Ilyashenko 1991] are not themselves oscillatory is insufficient. We give an explicit counterexample and we draw confines to which Ilyashenko's result may be restricted in order to keep the validity.
Paper Structure (15 sections, 2 theorems, 70 equations, 2 figures)

This paper contains 15 sections, 2 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The known case and the main difficulty
  3. Hyperbolic polycycles
  4. Adding more depth
  5. Current state of the proofs
  6. Commonalities
  7. Ecalle's approach
  8. Ilyashenko's approach
  9. Goals
  10. Splitting up the return map and the logarithmic chart
  11. Strategy of proof
  12. Sketch of definition of $FC^{1}$ and ordering of asympotics
  13. The problem and a counterexample
  14. Statement \ref{['StatementPos']}
  15. Bibliography

Key Result

Theorem 4.1

Let $A$ stand for conjugation by $\exp$, i.e.: All Dulac maps of the polycycles we study are contained in:

Figures (2)

  • Figure 1: Decomposition of a first return map of a simple alternant polycycle. The arrows on the separatrices are double for a hyperbolic separatrix and single for a center separatrix.
  • Figure 2: Counterexample notation

Theorems & Definitions (16)

  • Conjecture 1.1
  • Remark 2.3
  • Definition 3.1
  • Definition 3.2
  • Theorem 4.1: Structural Theorem, specific case
  • Remark 4.2
  • Remark 4.3
  • Definition 4.4
  • Remark 4.5
  • Remark 4.6
  • ...and 6 more