| Proceedings of the Second Isaac Congress: Volume 2: This Project Has Been Executed with Grant No. 11-56 from the Commemorative Association for the Jap Softcover Repri Edition Contributor(s): Begehr, Heinrich G. W. (Editor), Gilbert, R. P. (Editor), Kajiwara, Joji (Editor) |
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ISBN: 1461379717 ISBN-13: 9781461379713 Publisher: Springer OUR PRICE: $208.99 Product Type: Paperback - Other Formats Published: September 2011 |
| Additional Information |
| BISAC Categories: - Mathematics | Differential Equations - General - Mathematics | Mathematical Analysis - Mathematics | Functional Analysis |
| Dewey: 515 |
| Series: International Society for Analysis, Applications and Computa |
| Physical Information: 1.67" H x 6.14" W x 9.21" (2.54 lbs) 821 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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, |