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A Fundamental Domain for $$V_3$$
 Mémoires de la Société Mathématique de France 2010; 139 pp; softcover Number: 121 ISBN-10: 2-85629-301-8 ISBN-13: 978-2-85629-301-0 List Price: US$42 Member Price: US$33.60 Order Code: SMFMEM/121 The author describes a fundamental domain for the punctured Riemann surface $$V_{3,m}$$ which parametrises (up to Möbius conjugacy) the set of quadratic rational maps with numbered critical points, such that the first critical point has period three and the second critical point is not mapped in $$m$$ iterates or less to the periodic orbit of the first. This gives, in turn, a description, up to topological conjugacy, of all dynamics in all type III hyperbolic components in $$V_{3}$$, and gives indications of a topological model for $$V_{3}$$, together with the hyperbolic components contained in it. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in pure mathematics. Table of Contents Introduction The space $$V_3$$ Captures and counting The resident's view Fundamental domains Easy cases of the main theorem The hard case: Final statement and examples Proof of the hard and interesting case Open questions Bibliography