On the Asymptotic Behavior of the Trajectories of Semigroups of Holomorphic Functions
On the Asymptotic Behavior of the Trajectories of Semigroups of Holomorphic Functions
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关于全纯函数半群轨迹的渐近行为
DOI：
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发表时间：
2016
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通讯作者：
Dimitrios Betsakos
中科院分区：
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作者：
Dimitrios Betsakos
设$${phi _t}_{tge 0}$${ ϕtt}≥0为单位盘的全纯自映射的半群。我们设半群的Denjoy-Wolff点为点1；所以1是半群的唯一吸引边界不动点。我们进一步假设，对于所有$$tge 0$$ t≥0,$$phi _t^prime (1)=1$$ ϕt '(1)=1（角导数），即半群是抛物线型的。我们证明了Contreras和Díaz-Madrigal关于轨迹$$gamma _z(t)=phi _t(z)$$ γz(t)=ϕt(z)的渐近行为的猜想，当$$t
ightarrow +infty $$ t→+∞时。我们还证明了如果相关平面区域的边界包含在半条状区域中，则该半群的所有轨迹径向收敛于1。
Let $${phi _t}_{tge 0}$${ϕt}t≥0 be a semigroup of holomorphic self-maps of the unit disk. We assume that the Denjoy–Wolff point of the semigroup is the point 1; so 1 is the unique attractive boundary fixed point of the semigroup. We further assume that for all $$tge 0$$t≥0, $$phi _t^prime (1)=1$$ϕt′(1)=1 (angular derivative), namely the semigroup is parabolic. We disprove a conjecture of Contreras and Díaz-Madrigal on the asymptotic behavior of the trajectories $$gamma _z(t)=phi _t(z)$$γz(t)=ϕt(z), as $$t
ightarrow +infty $$t→+∞. We also prove that if the boundary of the associated planar domain is contained in a half-strip, then all the trajectories of the semigroup converge to 1 radially.