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 $$ \mathrm{Orb}(f,t_a)\triangleq \left\{ f^n:n \in \mathbb{N} \right\}\qquad f^0\triangleq f, $$ to be dense in $H(\mathbb{C})$?

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 $H(\mathbb{C})$?

Let $H(\mathbb{C})$ be the set 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; ie for $$ Orb(f,t_a)\triangleq \left\{ f^n:n \in \mathbb{N} \right\}\qquad f^0\triangleq f, $$ to be dense in $H(\mathbb{C})$?

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 $$ \mathrm{Orb}(f,t_a)\triangleq \left\{ f^n:n \in \mathbb{N} \right\}\qquad f^0\triangleq f, $$ to be dense in $H(\mathbb{C})$?