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A note on Bohr chaos and hyperbolic sets

Noriaki Kawaguchi

Abstract

This paper studies the relationship between shadowing phenomena and Bohr chaos in dynamical systems. We provide sufficient conditions for Bohr chaos in terms of shadowing. By combining those conditions with the shadowing lemma, we obtain some results on Bohr chaos and hyperbolic sets. Our results also highlight some simple but non-trivial structural properties of hyperbolic sets.

A note on Bohr chaos and hyperbolic sets

Abstract

This paper studies the relationship between shadowing phenomena and Bohr chaos in dynamical systems. We provide sufficient conditions for Bohr chaos in terms of shadowing. By combining those conditions with the shadowing lemma, we obtain some results on Bohr chaos and hyperbolic sets. Our results also highlight some simple but non-trivial structural properties of hyperbolic sets.
Paper Structure (9 theorems, 48 equations)

This paper contains 9 theorems, 48 equations.

Key Result

Theorem 1

Given a homeomorphism $f\colon X\to X$, if there are $x,y\in X$ with $x\ne y$ and a closed subset $S$ of $X$ with $f(S)=S$ such that then $f$ is Bohr chaotic.

Theorems & Definitions (18)

  • Remark 1
  • Definition 1
  • Theorem 1
  • proof : Proof of Theorem 1
  • Corollary 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Lemma 2
  • ...and 8 more