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Let $H(\mathbb{C})$ be the space of holomorphic functions on the complex plane. Then it is well-known that for $a\neq 0$, the translation operator $$ t_a(f)\triangleq f(x)\mapsto f(x+a), $$ is topologically transitive on $H(\mathbb{C})$. Are there known, sufficient conditions for $f$ to by a cyclic vector of this map; i.e. for $$ \mathrm{Orb}(f,t_a)\triangleq \left\{ t_a^n(f):n \in \mathbb{N} \right\}$$ to be dense in $H(\mathbb{C})$?

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    $\begingroup$ Sch functions are called hypercyclic (for the translation). $\endgroup$
    – Jochen Wengenroth
    Commented Sep 30, 2019 at 7:52
  • $\begingroup$ This I do know, but I can't find a characterization of them... $\endgroup$
    – ABIM
    Commented Sep 30, 2019 at 7:53

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Such functions are called universal entire functions. Actually most entire functions have this property. For specific examples, $\zeta$ function has this property.

MR0771576 Duĭos Ruis, Universal functions and the structure of the space of entire functions. (Russian) Dokl. Akad. Nauk SSSR 279 (1984), no. 4, 792–795.

Laurinčikas, A. The universality of zeta-functions. (English summary) Proceedings of the Eighth Vilnius Conference on Probability Theory and Mathematical Statistics, Part I (2002). Acta Appl. Math. 78 (2003), no. 1-3, 251–271.

MR3058520 Gauthier, Paul M. Approximating all meromorphic functions by linear motions of the Riemann zeta-function. Comput. Methods Funct. Theory 12 (2012), no. 2, 517–526.

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  • $\begingroup$ Do you know of a translation/summary of the main results of the first paper? $\endgroup$
    – ABIM
    Commented Sep 30, 2019 at 2:27
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    $\begingroup$ I read the review from MR. He proves that the set of universal functions is residual (the complement has first Baire's category). He also proves a similar result with translation replaced by differentiation. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 30, 2019 at 12:56
  • $\begingroup$ Oh, in this case I do know of such a result. I've seen it in a book on hypercylic operators, but I have not seen the other two :) Thanks. $\endgroup$
    – ABIM
    Commented Sep 30, 2019 at 13:02
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    $\begingroup$ Zbl says that there is an English translation: Sov. Math., Dokl. 30, 713-716 (1984) $\endgroup$
    – Alexandre Eremenko
    Commented Sep 30, 2019 at 13:09
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    $\begingroup$ If you type "universal entire function" in MR or Zbl you obtain about 100 results. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 30, 2019 at 13:10

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