Iteration of certain meromorphic functions with unbounded singular values

作者：Nayak Tarakanta^*; Prasad M Guru Prem

来源：Ergodic Theory and Dynamical Systems, 2010, 30(03): 877-891.

DOI：10.1017/S0143385709000364

Let M = {f (z)= (z(m)/sinh(m) z) for z is an element of C vertical bar either in or m//2 is an odd natural number) For each f is an element of M. the set of singularities of the inverse function of f is an unbounded subset of the real line R. In this paper, the iteration of functions in oneparameter family S = {f(lambda)(z) = lambda f (z) {lambda is an element of R \ {0}} is investigated for each f is an element of M. It is shown that, for each f is an element of M, there is a critical parameter lambda* > 0 dependine on f such that a period-doubling bifurcation occurs in the dynamics of functions f(lambda) in S when the parameter vertical bar lambda vertical bar passes through lambda* The non-existence of Baker domains and wandering domains in the Fatou set of f(lambda) is proved. Further, it is shown that the Fatou set of IA is infinitely connected for 0 < vertical bar lambda vertical bar <= lambda* whereas for vertical bar lambda vertical bar <= lambda*, the Fatou set of f(lambda) consists of infinitely many components and each component is simply connected.

• 出版日期2010-6