Abstract
In this article we answer in the negative the question of whether hypercyclicity is sufficient for distributional chaos for a continuous linear operator (we even prove that the mixing property does not suffice). Moreover, we show that an extremal situation is possible: There are (hypercyclic and non-hypercyclic) operators such that the whole space consists, except zero, of distributionally irregular vectors.
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Acknowledgments
The research of first and third author was supported by MEC and FEDER, project MTM2010-14909 and by GV, Project PROMETEO/2008/101. The research of second author was supported by the Marie Curie European Reintegration Grant of the European Commission under grant agreement no. PERG08-GA-2010-272297. The financial support of these institutions is hereby gratefully acknowledged. We also want to thank X. Barrachina for pointing out to us a gap in the proof of a previous version of Theorem 3.1.
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Martínez-Giménez, F., Oprocha, P. & Peris, A. Distributional chaos for operators with full scrambled sets. Math. Z. 274, 603–612 (2013). https://doi.org/10.1007/s00209-012-1087-8
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DOI: https://doi.org/10.1007/s00209-012-1087-8