On a Theorem Of
نویسنده
چکیده
This is the summary of a more detailed study on transformation groups on a metric phase space X, which is equicontinuous except for a set N(X) which is topologically sufficiently small compared to its complement. Kerékjartó [7] (1934) gave the following results. Let ƒ be an orientation preserving self-homeomorphism of the 2-sphere S whose powers (both positive and negative) are equicontinuous except for a finite number of points. Then (A) The number of exceptional points is at most 2. (B) For the no exceptional point case, ƒ is topologically conjugated to a rotation about a diameter; for the one exceptional point case, ƒ is topologically conjugated to the homeomorphism obtained from extending the translation on the plane by adding the fixed point { °° } ; for the two exceptional points case, ƒ is topologically conjugated to the homeomorphism obtained from extending the dilatation on the plane by adding the fixed point { °° }. In this paper we, among other things, obtain an extension of previous works on generalization of (A) by Homma-Kinoshita [5], Kaul [ó], and Gray-Roberson [4]. For the difficulty of extending (B) to more general cases consult Montgomery-Zippen [10, p. 229] and Kinoshita [8]. We assume in this announcement that (X, T, ir) is a transformation group [3] and X is a locally compact metric space with metric d.
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