In this paper, we study the $${\mathcal {F}}$$ -transitive behaviour of the translation semigroups on complex sectors, where $${\mathcal {F}}$$ is a Furstenberg family of the semigroup. We present the characterizations for $${\mathcal {F}}$$ -transitive translation semigroups on complex sectors, which implies that the $${\mathcal {F}}$$ -transitivity equals the thickly $${\mathcal {F}}$$ -transitivity for the translation semigroups on complex sectors. The relationship between the recurrence and the $${\mathcal {F}}$$ -transitivity for the translation semigroup is illustrated. The obtained characterizations generalize the recent results of Conejero and Peris (Discrete Contin Dyn Syst 25(4):1195–1208, 2009).

中文翻译：

---

复杂扇区上平移半群的 $${\mathcal {F}}$$-传递性和递归性

在本文中，我们研究了复杂扇区上平移半群的 $${\mathcal {F}}$$ -传递行为，其中 $${\mathcal {F}}$$ 是半群的 Furstenberg 族。我们给出了复杂扇区上 $${\mathcal {F}}$$ - 传递平移半群的表征，这意味着 $${\mathcal {F}}$$ - 传递性等于厚 $${\mathcal { F}}$$ - 复杂扇区上平移半群的传递性。说明了平移半群的递归和 $${\mathcal {F}}$$ - 传递性之间的关系。获得的特征概括了 Conejero 和 Peris 的最新结果（Discrete Contin Dyn Syst 25(4):1195–1208, 2009）。