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Strange distributionally chaotic triangular maps III

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  • Paganoni, L.
  • Smítal, J.

Abstract

In the class T of triangular maps of the square we consider the strongest notion of distributional chaos, DC1, originally introduced by Schweizer and Smítal [Trans Amer Math Soc 1994;344:737–854] for continuous maps of the interval. We show that a map F∈T is DC1 if F has a periodic orbit with period≠2n, for any n⩾0. Consequently, a map in T is DC1 if it has a homoclinic trajectory. This result is important since in general systems like T, positive topological entropy itself does not imply DC1. It contributes to the solution of a long-standing open problem of A. N. Sharkovsky concerning classification of triangular maps of the square.

Suggested Citation

  • Paganoni, L. & Smítal, J., 2008. "Strange distributionally chaotic triangular maps III," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 517-524.
    • Handle: RePEc:eee:chsofr:v:37:y:2008:i:2:p:517-524
    DOI: 10.1016/j.chaos.2006.09.037
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    References listed on IDEAS

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    1. Balibrea, F. & Smı́tal, J. & Štefánková, M., 2005. "The three versions of distributional chaos," Chaos, Solitons & Fractals, Elsevier, vol. 23(5), pages 1581-1583.
    2. Paganoni, L. & Smítal, J., 2005. "Strange distributionally chaotic triangular maps," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 581-589.
    3. Paganoni, L. & Smítal, J., 2006. "Strange distributionally chaotic triangular maps II," Chaos, Solitons & Fractals, Elsevier, vol. 28(5), pages 1356-1365.
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