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On exponentially accurate approximation of a near the identity map by an autonomous flow

V. Gelfreich, A. Vieiro

Abstract

This paper contains a proof of a refined version of Neishtadt's theorem which states that an analytic near-identity map can be approximated by the time-one map of an autonomous flow with exponential accuracy. We provide explicit expressions for the vector fields and give explicit bounds for the error terms.

On exponentially accurate approximation of a near the identity map by an autonomous flow

Abstract

This paper contains a proof of a refined version of Neishtadt's theorem which states that an analytic near-identity map can be approximated by the time-one map of an autonomous flow with exponential accuracy. We provide explicit expressions for the vector fields and give explicit bounds for the error terms.

Paper Structure

This paper contains 1 section, 1 theorem, 28 equations.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

If a map $f$ is analytic in $D_\delta$ and $\varepsilon/\delta \le 1/6e$, then the interpolating vector field $X_m$ of order $2\le m\le M_\varepsilon+1$, where $M_\varepsilon=\frac{\delta}{6\mathrm e\varepsilon}$, is analytic in $D_{\delta/3}$, $\|X_m\|_{D_{\delta/3}}\le 2\varepsilon$ and Moreover, for $m = \left\lfloor M_\varepsilon \right\rfloor+1$

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Remark 2
  • proof