Proceedings of the Second ISAAC Congress: Volume 2: This project has been executed with Grant No. 11–56 from the Commemorative Association for the ... Analysis, Applications and Computation, 8). Este artículo no está disponible.Proceedings of the Second ISAAC Congress: Volume 2: This project has been executed with Grant No. 11–56 from the Commemorative Association for the ... Analysis, Applications and Computation, 8)

Let 8 be a Riemann surface of analytically finite type (9, n) with 29 ­ 2+n> O. Take two pointsP1, P2 E 8, and set 8 ,1>2= 8 {P1’ P2}. Let PI Homeo+(8;P1,P2) be the group of all orientation preserving homeomor­ phismsw: 8 -+ 8 fixingP1, P2 and isotopic to the identity on 8. Denote byHomeot(8;Pb P2) the set of all elements ofHomeo+(8;P1, P2) iso­ topic to the identity on 8 ,P2’ ThenHomeot(8;P1,P2) is a normal sub­ pl group ofHomeo+(8;P1,P2). We setIsot(8;P1,P2) =Homeo+(8;P1,P2)/ Homeot(8;p1, P2). The purpose of this note is to announce a result on the Nielsen­ Thurston-Bers type classification of an element [w] ofIsot+(8;P1,P2). We give a necessary and sufficient condition for thetypeto be hyperbolic. The condition is described in terms of properties of the pure braid [b ] w induced by [w]. Proofs will appear elsewhere. The problem considered in this note and the form ofthe solution are suggested by Kra’s beautiful theorem in [6], where he treats self-maps of Riemann surfaces with one specified point. 2 TheclassificationduetoBers Let us recall the classification of elements of the mapping class group due to Bers (see Bers [1]). LetT(R) be the Teichmiiller space of a Riemann surfaceR, andMod(R) be the Teichmtiller modular group of R. Note that an orientation preserving homeomorphism w: R -+ R induces canonically an element (w) EMod(R). Denote by&.r(R)(·,.) the Teichmiiller distance onT(R). For an elementXEMod(R), we define a(x)= inf &.r(R)(r,x(r)).

Let 8 be a Riemann surface of analytically finite type (9, n) with 29 ­ 2+n> O. Take two pointsP1, P2 E 8, and set 8 ,1>2= 8 \ {P1' P2}. Let PI Homeo+(8;P1,P2) be the group of all orientation preserving homeomor­ phismsw: 8 -+ 8 fixingP1, P2 and isotopic to the identity on 8. Denote byHomeot(8;Pb P2) the set of all elements ofHomeo+(8;P1, P2) iso­ topic to the identity on 8 ,P2' ThenHomeot(8;P1,P2) is a normal sub­ pl group ofHomeo+(8;P1,P2). We setIsot(8;P1,P2) =Homeo+(8;P1,P2)/ Homeot(8;p1, P2). The purpose of this note is to announce a result on the Nielsen­ Thurston-Bers type classification of an element [w] ofIsot+(8;P1,P2). We give a necessary and sufficient condition for thetypeto be hyperbolic. The condition is described in terms of properties of the pure braid [b ] w induced by [w]. Proofs will appear elsewhere. The problem considered in this note and the form ofthe solution are suggested by Kra's beautiful theorem in [6], where he treats self-maps of Riemann surfaces with one specified point. 2 TheclassificationduetoBers Let us recall the classification of elements of the mapping class group due to Bers (see Bers [1]). LetT(R) be the Teichmiiller space of a Riemann surfaceR, andMod(R) be the Teichmtiller modular group of R. Note that an orientation preserving homeomorphism w: R -+ R induces canonically an element (w) EMod(R). Denote by&.r(R)(·,.) the Teichmiiller distance onT(R). For an elementXEMod(R), we define a(x)= inf &.r(R)(r,x(r)).