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Topological entropy for induced hyperspace maps

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  • Cánovas Peña, Jose S.
  • López, Gabriel Soler

Abstract

Let (X,d) be a compact metric space and let f:X→X be continuous. Let K(X) be the family of compact subsets of X endowed with the Hausdorff metric and define the extension f¯:K(X)→K(X) by f¯(K)=f(K) for any K∈K(X). We prove that the topological entropy of f¯ is greater or equal than the topological entropy of f, and this inequality can be strict. On the other hand, we prove that the topological entropy of f is positive if and only if the topological entropy of f¯ is also positive.

Suggested Citation

  • Cánovas Peña, Jose S. & López, Gabriel Soler, 2006. "Topological entropy for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 979-982.
    • Handle: RePEc:eee:chsofr:v:28:y:2006:i:4:p:979-982
    DOI: 10.1016/j.chaos.2005.08.173
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    References listed on IDEAS

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    1. Peris, Alfredo, 2005. "Set-valued discrete chaos," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 19-23.
    2. Román-Flores, Heriberto & Chalco-Cano, Y., 2005. "Robinson’s chaos in set-valued discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 33-42.
    3. Banks, John, 2005. "Chaos for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 681-685.
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