Distributional chaos for operators on Banach spaces

By Nilson C. Bernardes, Antonio Bonilla, Alfredo Peris, Xinxing Wu. 2018

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Abstract Four notions of distributional chaos, namely DC1, DC2, DC 2 1 2 and DC3, are studied within the framework of operators on Banach spaces. It is known that, for general dynamical systems, DC1 ⊂ DC2 ⊂ DC 2 1 2 ⊂ DC3. We show that DC1 and DC2 coincide in our context, which answers a natural question. In contrast, there exist DC 2 1 2 operators which are not DC2. Under the condition that there exists a dense set X 0 ⊂ X such that T n x → 0 for any x ∈ X 0 , DC3 operators are shown to be DC1. Moreover, we prove that any upper-frequently hypercyclic operator is DC 2 1 2 . Finally, several examples are provided to distinguish between different notions of distributional chaos, Li–Yorke chaos and irregularity.

Subfields: Applied Mathematics, Mathematical Physics

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