• Title/Summary/Keyword: robustly transitive set

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A PERSISTENTLY SINGULAR MAP OF 𝕋n THAT IS C2 ROBUSTLY TRANSITIVE BUT IS NOT C1 ROBUSTLY TRANSITIVE

  • Morelli, Juan Carlos
    • Journal of the Korean Mathematical Society
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    • v.58 no.4
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    • pp.977-1000
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    • 2021
  • Consider the high dimensional torus 𝕋n and the set 𝜺 of its endomorphisms. We construct a map in 𝜺 that is robustly transitive if 𝜺 is endowed with the C2 topology but is not robustly transitive if 𝜺 is endowed with the C1 topology.

HYPERBOLIC STRUCTURE OF POINTWISE INVERSE PSEUDO-ORBIT TRACING PROPERTY FOR C1 DIFFEOMORPHISMS

  • Manseob Lee
    • Communications of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.243-256
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    • 2023
  • We deal with a type of inverse pseudo-orbit tracing property with respect to the class of continuous methods, as suggested and studied by Pilyugin [54]. In this paper, we consider a continuous method induced through the diffeomorphism of a compact smooth manifold, and using the concept, we proved the following: (i) If a diffeomorphism f of a compact smooth manifold M has the robustly pointwise inverse pseudoorbit tracing property, f is structurally stable. (ii) For a C1 generic diffeomorphism f of a compact smooth manifold M, if f has the pointwise inverse pseudo-orbit tracing property, f is structurally stable. (iii) If a diffeomorphism f has the robustly pointwise inverse pseudo-orbit tracing property around a transitive set Λ, then Λ is hyperbolic for f. Finally, (iv) for C1 generically, if a diffeomorphism f has the pointwise inverse pseudo-orbit tracing property around a locally maximal transitive set Λ, then Λ is hyperbolic for f. In addition, we investigate cases of volume preserving diffeomorphisms.