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Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1781–1800 | Cite as

Improbability of Wandering Orbits Passing Through a Sequence of Poincaré Surfaces of Decreasing Size

  • Stefan Fleischer
  • Andreas KnaufEmail author
Article

Abstract

Given a volume preserving dynamical system with non-compact phase space, one is sometimes interested in special subsets of its wandering set. One example from celestial mechanics is the set of initial values leading to collision. Another one is the set of initial values of semi-orbits, whose asymptotic velocity does not exist as a limit. We introduce techniques that can be helpful in showing that these sets are of measure zero. We do this by defining a sequence of hypersurfaces, that are eventually hit by each of those semi-orbits and whose total surface area decreases to zero.

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Notes

Acknowledgements

We thank the anonymous referee for helpful comments and suggestions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsFriedrich-Alexander-University Erlangen-NürnbergErlangenGermany

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