• 제목/요약/키워드: continuum-wise expansiveness

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CONTINUUM-WISE EXPANSIVENESS FOR C1 GENERIC VECTOR FIELDS

  • Manseob Lee
    • 대한수학회지
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    • 제60권5호
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    • pp.987-998
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    • 2023
  • It is shown that every continuum-wise expansive C1 generic vector field X on a compact connected smooth manifold M satisfies Axiom A and has no cycles, and every continuum-wise expansive homoclinic class of a C1 generic vector field X on a compact connected smooth manifold M is hyperbolic. Moreover, every continuum-wise expansive C1 generic divergence-free vector field X on a compact connected smooth manifold M is Anosov.

UNIVERSALLY MEASURE CONTINUUM-WISE EXPANSIVE HOMOCLINIC CLASSES

  • Daejung Kim;Seunghee Lee;Junmi Park
    • 충청수학회지
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    • 제36권3호
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    • pp.171-180
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    • 2023
  • Investigating local dynamics requires precise control to effectively manage the subtle differences that distinguish it from global dynamics. This paper aims to study the localized perspective of the recently proposed continuum-wise expansive measures [13]. Let f : M → M be a diffeomorphism on a closed smooth manifold M and let p be a hyperbolic periodic point of f. We prove that if the homoclinic class Hf (p) of f associated to p is C1-robustly measure continuum-wise expansive then it is hyperbolic.

PRESERVATION OF EXPANSIVITY IN HYPERSPACE DYNAMICAL SYSTEMS

  • Koo, Namjip;Lee, Hyunhee
    • 대한수학회지
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    • 제58권6호
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    • pp.1421-1431
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    • 2021
  • In this paper we study the preservation of various notions of expansivity in discrete dynamical systems and the induced map for n-fold symmetric products and hyperspaces. Then we give a characterization of a compact metric space admitting hyper N-expansive homeomorphisms via the topological dimension. More precisely, we show that C0-generically, any homeomorphism on a compact manifold is not hyper N-expansive for any N ∈ ℕ. Also we give some examples to illustrate our results.