Parabolic type semigroups: asymptotics and order of contact

Abstract

We study the asymptotic behavior of parabolic type semigroups acting on the unit disk as well as those acting on the right half-plane. We use the asymptotic behavior to investigate the local geometry of the semigroup trajectories near the boundary Denjoy–Wolff point. The geometric content includes, in particular, the asymptotes to trajectories, the so-called limit curvature, the order of contact, and so on. We then establish asymptotic rigidity properties for a broad class of semigroups of parabolic type.

Appendix

We complete our analysis with assertions which give more information about the asymptotic behavior of semigroups but are different in nature.

Proposition 5.1

Let $\Sigma =\{\Phi _{t}\}_{t\ge 0}\subset \hbox {Hol}(\Pi )$ be a semigroup generated by $\phi \in \mathcal {G}_{\alpha ,\beta }(\Pi )$. Then

$$\begin{aligned} \lim _{t\rightarrow \infty } (t+1)^{\frac{\beta }{\alpha }} \left( (\Phi _t(w)+1)^{\alpha }-(\Phi _t(1)+1)^{\alpha } - \lambda \sigma (w)\right) =\mu \lambda ^{1-\frac{\beta }{\alpha }}\sigma (w). \end{aligned}$$

Proof

We just calculate the limit:

$$\begin{aligned}&\lim _{t\rightarrow \infty } (t+1)^{\frac{\beta }{\alpha }}\left( (\Phi _t+1)^{\alpha }(w)-(\Phi _t+1)^{\alpha }(1) -\lambda \sigma (w)\right) \\&\quad =\lim _{t\rightarrow \infty } (t+1)^{\frac{\beta }{\alpha }}\cdot \int \limits _1^w \left( (\Phi _t(z)+1)^{\alpha }-\lambda \sigma (z)\right) 'dz \\&\quad = \lim _{t\rightarrow \infty } (t+1)^{\frac{\beta }{\alpha }}\cdot \int \limits _1^w \frac{\alpha (\Phi _t(z)+1)^{\alpha -1}\phi \left( \Phi _t(z)\right) -\lambda }{\phi (z)} dz. \end{aligned}$$

Since

$$\begin{aligned}&\lim _{t\rightarrow \infty }(t+1)^{\frac{\beta }{\alpha }} \left( \alpha (\Phi _t(z)+1)^{\alpha -1}\phi \left( \Phi _t(z)\right) -\lambda \right) \\&\quad =\lim _{t\rightarrow \infty }\left( \frac{(t+1)^{\frac{1}{\alpha }}}{(\Phi _t(z)+1)}\right) ^{\beta } \left( \alpha B+\frac{\rho _{1}((\Phi _t(z))}{(\Phi _t(z)+1)^{1-\alpha -\beta }}\right) =\mu \lambda ^{1-\frac{\beta }{\alpha }}, \end{aligned}$$

we conclude that

$$\begin{aligned} \lim _{t\rightarrow \infty } (t+1)^{\frac{\beta }{\alpha }}\left( (\Phi _t(w)+1)^{\alpha }-(\Phi _t(1)+1)^{\alpha } - \lambda \sigma (w)\right) =\mu \lambda ^{1-\frac{\beta }{\alpha }}\sigma (w), \end{aligned}$$

which completes the proof. $\square$

The particular case $\alpha =\beta =1$ is contained in [10, Theorem 4.1(ii)]. Transferring, as above, Proposition 5.1 to semigroups acting in $\Delta$ yields the following result.

Corollary 5.1

Let $S=\left\{ F_t\right\} _{t\ge 0}$ be a semigroup of holomorphic self-mappings of $\Delta$ generated by $f \in \mathcal {G}_{\alpha ,\beta }(\Delta )$. Then

$$\begin{aligned} \lim _{t\rightarrow \infty } (t+1)^{\frac{\beta }{\alpha }} \left( \frac{1}{(1-F_t(z))^\alpha } - \frac{1}{(1-F_t(0))^\alpha } - \frac{\lambda h(z)}{2^\alpha }\right) =\frac{\mu \lambda ^{1-\frac{\beta }{\alpha }} h(z)}{2^\alpha }. \end{aligned}$$