Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations

Abstract

By introducing for monotone recurrence relations pseudo solutions, which are analogues of pseudo orbits of dynamical systems, we show that for general monotone recurrence relations the rotation set is closed, and each element in the rotation set is realized by a Birkhoff orbit. Moreover, if there is an orbit without rotation number, then the system has positive topological entropy, and we can construct orbits shadowing different rotation numbers.

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Acknowledgements

We are grateful to the referee for some suggestions which help to improve the exposition of the paper.

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Correspondence to Wen-Xin Qin.

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Supported by the National Natural Science Foundation of China (11771316, 11790274).

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Zhou, T., Qin, W. Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations. Math. Z. (2020). https://doi.org/10.1007/s00209-020-02574-w

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Keywords

  • Rotation set
  • Pseudo solution
  • Topological entropy
  • Shadowing rotation
  • Monotone recurrence relation

Mathematics Subject Classification

  • 37B40
  • 37C65
  • 37E40
  • 37E45
  • 37L60