Abstract. In this project, we propose to study some Diophantine problems arising from the iterations of a rational function in one variable. Let K be a number eld (or a global eld in general). and let ' : P1 ! P1 be a rational map de ned over K with d = deg '(z) 2: Let 2 P1(K) be a xed rational point which is not a totally rami ed point of '2. Let " > 0 be given and let 2 P1(K) such that its '-orbit is ini nite. For any nite subset S MK containing all the archimedean places of K; the set ô€€€';S( ; ; ") of quasi-(S; ")-integral points with respect to in O'( ) is a nite set [25]. We [14] obtained a bound for the cardinality of this nite set. It is expected that there exists a uniform bound for the cardinality of the set of quasi-(S; ")-integral points provided that ' satis es some suitable minimal conditions. In this project, we propose to test such uniform bound and other related questions for the two (one parameter) quadratic families : 'c(z) = z2 + c and c(z) = z + c=z where c 2 K is a parameter for the family. There are several related questions that need to be studied: (a) minimal conditions (minimal models) for ', (b) lower bounds for the canonical heights of points in P1(K) whose '-orbit is in nite and (c) the dependence on the parameter 2 P1(K): Related to these two families, we'll also study the dynatomic curves Y dyn 1 (n) arising from the n- periodic points of formal period n: These curves are de ned by n(c; z) = 0 which are equations for the periodic points of formal periodi equal to n: In paritcular, we'll invetigate the group of cuspidal divisors and correspondingly, the group of dynmaical units of the coordinate ring K[Y dyn 1 (n)] of Y dyn 1 (n): ; 研究期間 9908 ~ 10007
