Growth of frequently or log-frequently hypercyclic functions
Let \(H(\mathbb{D})\) denote the space analytic functions on the open unit disc equipped with the standard topology of uniform convergence on compact sets. The Taylor shift operator \(T\) on \(H(\mathbb{D})\) is defined as follows for \(z\in\mathbb{D}\) and \(f(z)=\sum_{k\geq0}a_kz^k\): \[ T(f)(z)=T\left(\sum_{k\geq0}a_kz^k\right)=\sum_{k\geq0}a_{k+1}z^k. \] The dynamical properties of this operator (and many others) have been intensively studied in recent years. One of the most studied dynamical notion is that of frequent hypercyclicity introduced in [\textit{F. Bayart} and \textit{S. Grivaux}, Trans. Am. Math. Soc. 358, No. 11, 5083--5117 (2006; Zbl 1115.47005)]. On an infinite-dimensional Fréchet space \(F\), an operator \(T\) is called frequently hypercyclic if there exists a vector \(x\in F\) (also called frequently hypercyclic) such that for any non-empty open set \(U\subseteq F\), the classical lower density of the return sets \(N(x,U):=\{n\in\mathbb{N}:T^nx\in U\}\) is positive where the lower density of a subset \(A\subseteq\mathbb{N}\) is defined as \[ \underline{d}(A):=\liminf_{N\to\infty}\frac{\#\left(A\cap[0;N]\right)}{N+1}. \] One may also define a weaker notion of log-frequent hypercyclicity by replacing the lower density in the previous definition by the so-called log-density \(\underline{d}_{log}\) given by \[ \underline{d}_{log}(A):=\liminf_{N\to\infty}\frac{1}{\log(N)}\sum_{n\in A\cap[0;N]}\frac{1}{n}. \] In particular, when dealing with frequently hypercyclic operators on spaces of analytic functions, a classical subject of interest is the rate of growth of frequently hypercyclic functions. Recall that for \(f\in H(\mathbb{D})\) and \(1\leq p<\infty\), \[ M_p(r,f):=\left(\frac{1}{2\pi}\int_{0}^{2\pi}\left\vert f(re^{i\theta})\right\vert^p d\theta\right)^{\frac{1}{p}}\text{ and }M_{\infty}(r,f):=\sup_{\theta\in[0,2\pi]}{\left\vert f(re^{i\theta})\right\vert}. \] The rate of growth of frequently hypercyclic functions for the Taylor shift on \(H(\mathbb{D})\) has already been studied by the authors [J. Anal. Math. 143, No. 2, 615--637 (2021; Zbl 07382545)] in terms of average \(L^p\)-norms for \(p\in[1,\infty]\) and for \(1<p\leq \infty\) (and \(q\) the conjugate exponent) where they proved that there exists a frequently hypercyclic function \(f\) for \(T\) on \(H(\mathbb{D})\) and \(C>0\) such that, for every \(0<r<1\), \[ M_p(r,f)\leq \frac{C}{(1-r)^{\frac{1}{\max(2,q)}}}. \] Moreover, they complement this study by proving that this rate is optimal since every frequently hypercyclic function \(f\in H(\mathbb{D})\) satisfies \[ \liminf_{r\to1}(1-r)^{\frac{1}{\max(2,q)}}M_p(r,f)>0. \] In this article, the authors study the rate of growth of log-frequently hypercyclic functions for the Taylor shift. They give explicit formulae for these growths and prove that their results are optimal. It is worth noticing that, as one may expect, the range of growth of log-frequently hypercyclic functions is wider than that of frequently hypercyclic functions. Indeed, for \(1<p\leq \infty\), they prove that every log-frequently hypercyclic function satisfies \[ \limsup_{r\to1}(1-r)^{\frac{1}{\max(2,q)}}M_p(r,f)>0 \] and that there exists \(C\geq1\) such that \[ \liminf_{r\to1}(1-r)^{\frac{1}{\max(2C,qC)}}M_p(r,f)>0. \] They complement this result by proving that it is in a way optimal since for every large enough integer \(a\), there exists a log-frequently hypercyclic function \(f_a\in H(\mathbb{D})\) and a constant \(C>0\) such that \[ M_p(r,f_a)\leq \frac{C}{(1-r)^{\frac{1}{\max(2,q)}}} \] and \[ \liminf_{r\to1}(1-r)^{\frac{1}{\max(2a,qa)}}M_p(r,f)<\infty. \] They also study the case \(p=1\) which turns out have the same rate of growth as in the frequently hypercyclic function case. Finally, they turn to the study of the rate of growth of frequently hypercyclic functions for the differentiation operator on \(H(\mathbb{C})\) for which the best rate of growth is known [\textit{O. Blasco} et al., Proc. Edinb. Math. Soc., II. Ser. 53, No. 1, 39--59 (2010; Zbl 1230.47019); \textit{D. Drasin} and \textit{E. Saksman}, J. Funct. Anal. 263, No. 11, 3674--3688 (2012; Zbl 1315.47007)] and given by \[ \limsup_{r\to\infty}(r)^{\frac{1}{2\min(2,p)}}e^{-r}M_p(r,f_a)>0. \] The authors provide in this case the optimal rate for the lower bound of the growth of such functions.
scientific article; zbMATH DE number 7395024
scientific article