In this paper, we study the dynamics of the two-parameter family of rational maps $$\begin{eqnarray*}{F}_{a, b} (z)= {z}^{n} + \frac{a}{{z}^{n} } + b.\end{eqnarray*}$$ We give the topological description of Julia sets and Fatou components of ${F}_{a, b} $ according to the dynamical behavior of the orbits of its free critical points.

The rational maps no Herman ringsAbstract. It is proved that the rational maps in the family {z → z m + λ/z d : λ ∈ C\{0}} for integers m, d ≥ 2 have no Herman rings.

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In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

We prove that the Eisenstein-Picard modular group SU(2, 1; Z[ω 3 ]) can be generated by four given transformations.2010 Mathematics subject classification: primary 32M05; secondary 22E40, 32M15.