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Chain recurrence and average shadowing in dynamics

Abstract

We investigate several notions related to pseudotrajectories, including chain recurrence and shadowing properties, for a special class of diffeomorphisms on euclidean spheres, known as spherical linear transformations, and for bounded linear operators on Banach spaces. Our main results are complete characterizations of chain recurrence for spherical linear transformations on euclidean spheres and for weighted shifts on the classical Banach sequence spaces \(c_0\) and \(\ell _p\). Another main result is a characterization of hyperbolicity for invertible operators on Banach spaces by means of average expansivity and the average shadowing property.

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Acknowledgements

The authors thank the referees for their valuable suggestions. The second author was partially supported by project #304207/2018-7 of CNPq (National Council for Scientific and Technological Development - Brazil). The third author was partially supported by CNPq project 311018/2018-1, Fapesp project 2019/10269-3 and Capes print auxpe 88881.310741/2018-01.

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Correspondence to Nilson C. Bernardes Jr..

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Alves, F.F., Bernardes, N.C. & Messaoudi, A. Chain recurrence and average shadowing in dynamics. Monatsh Math (2021). https://doi.org/10.1007/s00605-021-01617-6

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Keywords

  • Pseudotrajectories
  • Chain recurrence
  • Shadowing properties
  • Spherical linear transformations
  • Linear operators
  • Weighted shifts

Mathematics Subject Classification

  • Primary 37B65
  • 37C50
  • Secondary 37B05
  • 37C05
  • 47A16