The three versions of distributional chaos

Strange distributionally chaotic triangular maps III

“…Recall that DC1 was originally introduced in [9] for the class C of continuous maps of the interval, for weaker notions DC2 and DC3 see, e.g., [11] or [2].…”

Distributional chaos for flows

“…For the discrete dynamical systems, both DC1 and DC2 are topological conjugacy invariants [5], but DC3 is not an invariant [1]. However, we will show that neither DC2 nor DC3 is an invariant for equivalent flows in Section 3.…”

“…see [1]) ϕ 1 is DC1 if and only if there are two points x, y ∈ M such that Ψ * xy (ε) = 1, ∀ε > 0 and Ψ xy (ε 0 ) = 0 for some ε 0 > 0. We first prove that for any ε > 0 there is a positive number ε 1 such that…”

“…The definitions of distributional chaos of flow follow some basic ideas of discrete systems (e.g. see [1], [4], [5]). In fact, we will prove, in Section 2 that both the distributional chaos DC1 and DC2 of a flow are equivalent to the time-1 maps and so some properties of DC1 and DC2 for discrete systems also hold for flows.…”

Chaos for Cowen-Douglas operators

“…Further results of [4] were extended in [13] to distributional chaos for the annihilation operator of a quantum harmonic oscillator. For more about distributional chaos, one can see [16,17,10,11,18].…”