Distributional chaos for linear operators

By Nilson C. Bernardes, Antonio Bonilla, Vladimír Müller, Alfredo Peris. 2013

105 citations recorded by the source corpus.

Abstract We characterize distributional chaos for linear operators on Frechet spaces in terms of a computable condition (DCC), and also as the existence of distributionally irregular vectors. A sufficient condition for the existence of dense uniformly distributionally irregular manifolds is presented, which is very general and can be applied to many classes of operators. Distributional chaos is also analyzed in connection with frequent hypercyclicity, and the particular cases of weighted shifts and composition operators are given as an illustration of the previous results.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e3c2ca56474c53a353b3abba0f060a410f277226

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Linear dynamical systems on Hilbert spaces: typical properties and explicit examples (2017, background): “There is no need to recall these definitions here, because things simplify greatly in the linear setting: as shown in [11], an operator T ∈ B(H) is distributionally chaotic if and only if it admits a distributionally irregular vector, i.”
Li–Yorke chaos in linear dynamics (2014, background): “A similar characterization of distributional chaos for operators on Fréchet spaces was recently obtained in [8].”
Frequently recurrent operators (2020, methodology): “(c) We use the same argument as in (b), based now on [8, Proposition 9].”
Shadowing and structural stability for operators (2020, background): “It simultaneously implies frequent hypercyclicity, mixing, Devaney chaos, dense distributional chaos and dense mean Li–Yorke chaos [7, 9, 16].”
Expansivity and Shadowing in Linear Dynamics (2016, background): “We refer the reader to the books [2, 19] and to the more recent papers [3, 6, 7, 8, 17, 18], where many additional references can be found.”
Difference sets and frequently hypercyclic weighted shifts (2013, background): “By [7], an operator with a distributionally unbounded orbit and such that T n x→ 0 for any x in a dense subset X0 of X is distributionally chaotic.”
Linear chaos and frequent hypercyclicity (2014, background): “In [10], the authors study the notion of distributionally chaotic operators and pose the following question.”
Invariant measures for frequently hypercyclic operators (2014, background): “In [9], the authors ask (a restricted form of) the following question.”