Classification of recurrent domains for some holomorphic maps

Abstract

abstract Riemann surface. As in Theorem 1.2 we show first that f is not constant on any component fl(,~v), l> 0. Hence we can assume using a diagonal process that for a subsequence (fm,),fm,~ h and h = ld on Ut~_of'(~_,p) c {q ~ g2; h(q) = q). Consequently U,~of'(~n) is closed in t2. Since fro, ~ ld on Zn and since f is not of finite order, see [FM], then f is conjugate to an irrational rotation on the Riemann surface E. We then show as in Theorem 2.3 of [FS3] that Z is contained in f2, so E is closed in I2. The rest of the argument is as in Theorem 1.2 or as in [FS3] Theorem 2.3. References [B] [BBD] [BS] [CG] [FM] [FS1] [FS2] [FS3] [FS4] [G] [HI [Ko] [Kr] [M] IN] [RR] [U] Beardon, A.: Iteration of rational functions. Springer Verlag (1991) Barrett, D., Bedford, E., Dadok, J.: T~-actions on holomorphically separable complex manifolds. Math. Z. 202 (1989), 65-82 Bedford, E., Smillie, J.: Polynomial diffeomorphisms of •2. II. Amer. J. Math. 4 (1991), 657-679 Carleson, L., Gamelin, T.: Complex dynamics. Springer Verlag (1993) Friedland, S., Milnor, J.: Dynamical properties of plane automorphisms. Ergodic theory and dynamical systems 9 (1989), 67-99 Fornaess, J.E., Sibony, N.: Complex dynamics in higher dimension. I. Asterisque 222 (1994), 201-231 Fornaess, J.E., Sibony, N.: Complex dynamics in higher dimension. II. To appear in Ann. Math. Studies Fornaess, J.E., Sibony, N.: Complex Henon mappings in 2 and Fatou-Bieberbach domains. Duke Math. J. 65 (1992), 345-380 Fornaess, J.E., Sibony, N.: Holomorphic dynamical systems. (To appear) Gavosto, E.: To appear Herman, M.: Recent results on some open questions on Siegel's linearization Kobayashi, S.: Hyperbolic manifolds and holomorphic mappings. New York, Marcel Dekker, 1970 Kruzhilin, N.G.: Holomorphic automorphisms of hyperbolic Reinhardt do- mains. Math. USSR Izvestya 32 (1989), 15-38 Milnor, J.: Dynamics in one complex variable: Introductory Lectures. SUNY Stony Brook. Institute for Mathematical Sciences. Preprint # 1990/5 Narashiman, R.: Several complex variables. Univ. Chicago Press, 1971 Rosay, J.-P., Rudin, W.: Holomorphic maps from IE n to C n. TAMS 310 (1988), 47-86 UEDA, T.: Fatou set in complex dynamics in projective spaces. Preprint