Fixed points and dynamics of two-parameter family of hyperbolic cosine like functions

Abstract

In this paper, the two-parameter family Fb≡{fλ,μ(z)=λbz+μb−zforz∈C:λ≥μ>0} of transcendental entire functions is considered. By investigating on the existence and nature of the fixed points of fλ,μ, the dynamics of functions fλ,μ∈Fb is studied. It is shown that the Julia set of fλ,μ is nowhere dense and not locally connected whenever parameters λ and μ satisfy 1+4λμ(ln⁡b)2≤ln⁡(1+1+4λμ(ln⁡b)22λln⁡b) and the Julia set is the whole of extended complex plane for 1+4λμ(ln⁡b)2>ln⁡(1+1+4λμ(ln⁡b)22λln⁡b).