Abstract

If $f$ is a holomorphic map, a Herman ring is defined to be an open set conformally equivalent to an annulus, on which some iterate of $f\; is\; conformally\; conjugate\; to\; a\; rigid\; rotation\; of\; some\; irrational\; angle.\; Periodic\; cycles\; of\; Herman\; rings\; were\; shown\; to\; exist\; for\; rational\; maps\; by\; Herman,\; and\; later\; by\; Shishikura,\; who\; also\; studied\; the\; possible\; configurations\; of\; these\; cycles\; on\; the\; complex\; plane.\; We\; show\; that\; periodic\; cycles\; of\; Herman\; rings\; also\; exist\; for\; transcendental\; meromorphic\; maps,\; and\; moreover,\; any\; configuration\; which\; is\; realizable\; by\; a\; rational\; map\; is\; also\; realizable\; by\; a\; transcendental\; meromorphic\; map.$