| Abstrakt: |
Dynamics of a one-parameter family of functions fλ(z)=λ+z+tanz,z∈Cand λ∈Cis investigated in this article. This function fλhas an unbounded set of singular values. The dynamics of fλ+mπis determined for all non-zero integers mand for the values of λsatisfying either |2+λ2|<1,or λ=i,or 2+λ2=e2πiαfor all rational numbers αand for each bounded type irrational number α.For such values of λ,the existence of |m| many wandering domains of fλ+mπwith disjoint grand orbits contained in the lower half-plane is asserted along with a completely invariant Baker domain containing the upper half-plane. Further, each such wandering domain is found to be simply connected, unbounded, and escaping. Different types of the internal behavior of {fλ+mπn}n>0on such a wandering domain Ware highlighted for different values of λ.More precisely, for ∣2+λ2∣<1,it is shown that the forward orbit of every point z∈Wstays away from the boundaries of Wns. For λ=i,it is proved that limn→∞dist(fi+mπn(z),∂Wn)=0for each z∈W.Further, ℑ(fi+mπn(z))→-∞as n→∞.For 2+λ2=e2πiαfor each rational number α,limn→∞dist(fλ+mπn(z),∂Wn)=0is established for each z∈W.But, ℑ(fλ+mπn(z))tends to a finite point for each z∈Wwhenever n→∞.For 2+λ2=e2πiα,limn→∞dist(fλ+mπn(z),∂Wn)>0for each z∈Wand for each bounded type irrational number α. |
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