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On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces with two specified points. (English) Zbl 1026.30037
Let $$S$$ be a hyperbolic Riemann surfaces $$R$$ of analytically finite type with two specific points $$p_1,p_2\in S$$, and set $$\dot S:=S\setminus \{p_1,p_2\}$$. Let $$I sot(S,2)$$ be the group of orientation preserving homeomorphisms of $$S$$ onto itself isotopic to $$id_S$$ and fixing the $$p_j$$ factored by the normal subgroup of homeomorphisms of $$S$$ isotopic to the identity of $$\dot S$$. Elements $$[\omega]\in I sot(S,2)$$ induce canonically elements $$\langle\omega |_S\rangle$$ of the Teichmüller modular group $$\text{Mod} (\dot S)$$. L. Bers [Acta Math. 141, 73-98 (1978; Zbl 0389.30018)] classified elements of $$\text{Mod}(\dot S)$$ as elliptic, parabolic and elliptic using the Teichmüller distance on the Teichmüller space $$T(\dot S)$$. In this paper the corresponding classification of elements $$[\omega]$$ of $$I sot(S,2)$$ is described using the strings of the induced pure braids $$[b_\omega]$$. The results are motivated by a theorem of I. Kra for surfaces with one specific point [Acta Math. 146, 231-270 (1981; Zbl 0477.32024)].

##### MSC:
 30F10 Compact Riemann surfaces and uniformization