Communications in Mathematical Physics

, Volume 80, Issue 4, pp 465–476 | Cite as

The baker transformation and a mapping associated to the restricted three body problem

  • Robert L. Devaney
Article

Abstract

The nonlinear mapping of the plane
$$\begin{array}{*{20}c} {x_1 = x_0 + 1/y_0 } \\ {y_1 = y_0 - x_0 - 1/y_0 } \\ \end{array} $$
was recently introduced by Hénon as an asymptotic form of the equations of motion of the restricted three body problem. This is an area preserving diffeomorphism, except along thex-axis where the mapping is singular. We show that this mapping exhibits a type of stochastic behavior known as topological transitivity, by showing that it is topologically conjugate to the well known baker transformation. Consequently, periodic points are dense in the plane and there is also a dense orbit. We note that the baker transformation also preserves Lebesgue measure and is ergodic, so this raises interesting open questions about the ergodic properties of the nonlinear mapping.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Lebesgue Measure 

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References

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Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Robert L. Devaney
    • 1
  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA

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