On connectivity numbers of stable domains
One aspect of complex iteration theory is the connectivity of stable domains. We prove the existence of infinitely connected preperiodic stable domains in a certain class of meromorphic. transcendental functions. AMS 1980 subject classification : 30D05 Let / : C —» Coo := C U {oc } be a meromorphic function and /" (η € No) the n—th iterate of / (/° := id). Then the Fatou set Τ and the Julia set J are defined as follows Τ := { ζ £ e » : (/n)n€Ν is defined and normal in some neighborhood of z}, J •= C o o \T. The components of Τ are called stable domains. A survey on the iteration theory of meromorphic functions may be found in [2]. We recall some elementary facts : Τ is completely invariant, i.e., } {J~) — /_1(JF) = T , if U is a stable domain, then there exists a stable domain Un with f»(U) ς un (i/o υ), if U and (/, are bounded in C then f(U) = Uu f(dU) = dU,. A stable domain U is called invariant if l\ = U, it is called periodic if Un = U for some n, and it. is called preperiodic if U is not periodic but Un = Um for some η > m > 1. In iteration theory the transcendental meromorphic functions are divided into several sub-