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Shadowing, asymptotic shadowing and s-limit shadowing

Volume 244 / 2019

Chris Good, Piotr Oprocha, Mate Puljiz Fundamenta Mathematicae 244 (2019), 287-312 MSC: Primary 37E05; Secondary 37C50, 37B10, 54H20, 26A18. DOI: 10.4064/fm492-5-2018 Published online: 21 December 2018

Abstract

We study three notions of shadowing: classical shadowing, limit (or asymptotic) shadowing, and s-limit shadowing. We show that classical and s-limit shadowing coincide for tent maps and, more generally, for piecewise linear interval maps with constant slopes, and are further equivalent to the linking property introduced by Chen in 1991.

We also construct a system which exhibits shadowing but not limit shadowing, and we study how shadowing properties transfer to maximal transitive subsystems and inverse limits (sometimes called natural extensions).

Where practicable, we show that our results are best possible by means of examples.

Authors

  • Chris GoodSchool of Mathematics
    University of Birmingham
    Birmingham, B15 2TT, UK
    e-mail
  • Piotr OprochaFaculty of Applied Mathematics
    AGH University of Science and Technology
    al. Mickiewicza 30
    30-059 Kraków, Poland
    and
    National Supercomputing Centre IT4Innovations
    Division of the University of Ostrava
    Institute for Research and Applications of
    Fuzzy Modeling
    30. Dubna 22
    70103 Ostrava, Czech Republic
    e-mail
  • Mate PuljizUniversity of Zagreb
    Faculty of Electrical Engineering
    and Computing
    Unska 3
    10000 Zagreb, Croatia
    e-mail
    e-mail

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