Existence and non-existence in common hypercyclicity and its frequent variants

Abstract

Given a family $\mathcal T = (T_λ)_{λ\inΛ}$ of bounded linear operators acting on the same $F$-space $X$ and indexed by a set of parameters $Λ\subset \mathbb{R}^d$, $d\geq1$, we explore the existence and non-existence of vectors that simultaneously satisfy, for all elements of $\mathcal T$, one of the following three dynamical properties: hypercyclicity, upper frequent hypercyclicity, and frequent hypercyclicity. Our approach relies on associating to $\mathcal T$ a "Lipschitz constant function'' $F$, whose growth encodes the interaction between the operators and the parameter set $Λ$. We show that the asymptotic behavior of $F$ determines sharp thresholds between existence and non-existence of common vectors. In the hypercyclicity setting, we identify the divergence of $\sum 1/F(n)^s$, where $s$ is the Hausdorff dimension of $Λ$, as a natural condition for existence, and prove complementary non-existence results, which are optimal for weighted shifts. For upper frequent hypercyclicity, we obtain criteria of non-existence in terms of series of the form $\sum 1/F(γ^n)$ for parameter sets in $d$-dimensions. Considered in the one-dimensional case, these criteria are optimal and allow us to contrast known existence results of Mestiri for logarithmic-type growth with non-existence beyond this scale. For frequent hypercyclicity, we establish a dichotomy showing that common vectors exist essentially only when $F$ is bounded, while any unbounded growth prevents their existence. Our results apply to a broad class of parameter sets, including self-similar fractals, providing a unified perspective on how growth conditions and geometric features of $Λ$ determine common dynamical behavior. Finally, in the specific context of weighted shifts, we prove that two operators do not necessarily share the same frequently hypercyclic vectors if the sequence of their weight products ratios admits two distinct non-zero cluster points, establishing the optimality of a result of Grivaux, Matheron and Menet.