A continuous map \(f: X\to X\) on a metric space \(X\) is said to be distributionally chaotic if there exist an uncountable set \(\Gamma\subseteq X\) and \(\varepsilon>0\) such that, for every \(\tau>0\) and \(x,y\in \Gamma\) with \(x\not=y\), we have \[ \underline{\mathrm{dens}} \{n\in\mathbb{N}: d(f^n(x),f^n(y))<\varepsilon\}=0, \;\overline{\mathrm{dens}} \{n\in\mathbb{N}: d(f^n(x),f^n(y))>\tau\}=1, \] where, for a set \(A\subseteq \mathbb{N}\), \(\underline{\mathrm{dens}} (A)\) and \(\overline{\mathrm{dens}} (A)\) denote the lower and upper densities of \(A\), respectively.
If \(X\) is a Fréchet space whose lc-topology is generated by the increasing sequence \((\|\cdot\|_k)_{k\in\mathbb{N}}\) of seminorms and \(T\) is a continuous linear operator on \(X\), a vector \(x\in X\) is said to be a distributionally irregular vector for \(T\) if there exist \(m\in\mathbb{N}\) and \(A, B\subseteq \mathbb{N}\) with \(\overline{\mathrm{dens}} (A)=\overline{\mathrm{dens}} (B)=1\) such that \(\lim_{n\in A}T^nx=0\) and \(\lim_{n\in B}\|T^n x\|_m=\infty\). The operator \(T\) is said to satisfy the Distributional Chaos Criterion (DCC) if there exist sequences \((x_k)\) and \((y_k)\) in \(X\) satisfying the following properties: there exists \(A\subseteq \mathbb{N}\) with \(\overline{\mathrm{dens}} (A)=1\) such that \(\lim_{n\in A}T^n x_k=0\) for all \(k\); \(y_k\in \overline{\text{span}\{x_n:\;n\in \mathbb{N}\}}\), \(\lim_{k\to\infty}y_k=0\), and there exists \(\varepsilon>0\) and an increasing sequence \((N_k)\) in \(\mathbb{N}\) such that \(\text{card}\{1\leq j\leq k: d(T^jy_k, 0)>\varepsilon\}\geq N_k(i-k^{-1})\) for all \(k\).
In the paper under review, the authors characterize distributional chaos for linear operators on Fréchet spaces in terms of the so-called condition (DCC) and also as the existence of distributionally irregular vectors. Precisely, they show that, if \(X\) is a Fréchet space and \(T\) is a continuous linear operator on \(X\), then \(T\) is distributionally chaotic if and only if \(T\) satisfies the (DCC) condition if and only if \(T\) has a distributionally irregular vector. A sufficient condition for the existence of dense uniformly distributionally irregular manifolds is also presented. Moreover, the authors characterize dense distributional chaos for unilateral weighted backward shifts on Fréchet spaces in terms of the existence of a distributionally unbounded orbit and for composition operators on the Fréchet space \(H({\mathbb{D}})\).