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A Discussion of Arnold's Limit Problem and its Geometric Argument

Keising Honn

Abstract

Upon re-examining Arnold's established lemma for explaining his famous limit problem, we have determined that while the lemma itself is correct, there is a defect in the original geometric proof. In this paper, we prove the correctness of the lemma using methods of power series, and construct a counterexample to illustrate the defect in Arnold's geometric proof.

A Discussion of Arnold's Limit Problem and its Geometric Argument

Abstract

Upon re-examining Arnold's established lemma for explaining his famous limit problem, we have determined that while the lemma itself is correct, there is a defect in the original geometric proof. In this paper, we prove the correctness of the lemma using methods of power series, and construct a counterexample to illustrate the defect in Arnold's geometric proof.

Paper Structure

This paper contains 7 sections, 3 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. An Analysis Argument of Problem (1) -- (2)
  3. Motivation to Detect a Defect
  4. Analysis argument of $|AB|/|BC| \rightarrow 1$.
  5. Analysis argument of $|ED|/|BC| \rightarrow 1$.
  6. Construction of the Counterexample
  7. Conclusions

Key Result

Lemma \oldthetheorem

If the graphs of analytic functions $f$ and $g$ do not coincide and they are both tangent to the line $y=x$ at the origin (Fig.fig: 1), then $|AB|/|BC|$ and $|BC|/|ED|$ converges to 1 as $A$ is sufficiently close to the origin.

Figures (3)

  • Figure 1: $f$ and $f^{-1}$, $g$ and $g^{-1}$ are symmetric about $y=x$ respectively
  • Figure 2: Convergence
  • Figure 3: Divergence

Theorems & Definitions (7)

  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • proof