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Fractional iteration and functional equations for functions analytic in the unit disk. (English) Zbl 1062.30026
Let $$\mathcal{P}$$ denote the set of all holomorphic functions $$f$$ in the open unit disk $$\mathbb D$$ with the condition $$f(\mathbb D) \subset \mathbb D$$. Then $$\mathcal{P}$$ is a topological semigroup with respect to the operation of composition and the topology of locally uniform convergence. Denote by $$\mathcal{I}$$ the set of all invertible elements of $$\mathcal{P}$$, i.e. $$\mathcal{I}$$ is the set of Möbius transformations which map the unit disk $$\mathbb D$$ onto itself. A function $$f \in \mathcal{P}$$ is called embeddable if there exists a family $$\{f^t\}_{t \geq 0}$$ in $$\mathcal{P}$$ such that $$f^0=\text{id}$$, $$f^1=f$$, $$f^{t+s} = f^t \circ f^s$$ for $$s$$, $$t \geq 0$$ and $$f^t \to \text{id}$$ as $$t \to 0+$$ locally uniformly in $$\mathbb D$$. Therefore, the map $$t \mapsto f^t$$ is a continuous homomorphism of the additive semigroup $$\mathbb R^+ = \{\, t \in \mathbb R : t \geq 0 \,\}$$ into $$\mathcal{P}$$, and thus $$\{f^t\}_{t \geq 0}$$ is a one-parameter continuous semigroup in $$\mathcal{P}$$. In this paper, the authors establish criteria for functions $$f \in \mathcal{P}$$ to be embeddable. Here, the Abel and Schröder functional equations play an important role.
To be more precise some further notation is necessary. Regarding iterates $$f^n$$, $$n \in \mathbb N$$ of $$f \in \mathcal{P}$$ as a dynamical system, one has to consider the nature of fixed points of $$f$$. In this connection an important role is played by the classical result of Denjoy and Wolff which asserts that for every $$f \in \mathcal{P} \setminus \mathcal{I}$$ there exists a unique point $$q$$, $$| q| \leq 1$$ such that $$f^n \to q$$ as $$n\to\infty$$ locally uniformly in $$\mathbb D$$. If $$q \in \mathbb D$$, then $$f(q)=q$$. In the case $$| q| =1$$, there exist the angular limits $$f(q) := \lim\limits_{z \to q}{f(z)}$$ and $$f'(q) := \lim\limits_{z \to q}{f'(z)}$$ with $$f(q)=q$$ and $$0 < f'(q) \leq 1$$. This point $$q$$ is called the Denjoy-Wolff point of $$f$$. It is also the Denjoy-Wolff point of all iterates $$f^n$$ of $$f$$. Therefore, it is natural to consider the subsemigroups $$\mathcal{P}[q]$$ of $$\mathcal{P}$$ which consist of all functions $$f \in \mathcal{P}$$ that have $$q$$ as their Denjoy-Wolff point. If $$f \in \mathcal{P}$$ and $$l \in \mathcal{I}$$, then $$f$$ is embeddable if and only if $$g = l \circ f \circ l^{-1}$$ is embeddable. In addition, every element in $$\mathcal{I}$$ is embeddable in a one-parameter group of automorphisms of $$\mathbb D$$. Therefore, it is sufficient to draw attention to $$\mathcal{P} \setminus \mathcal{I}$$ and $$\mathcal{P} \setminus \mathcal{I}$$. Now, we can state two of the author’s results.
Theorem 1. Let $$f \in \mathcal{P} \setminus \mathcal{I}$$, and let $$f'(0) = \gamma \neq 0$$. Then $$f$$ is embeddable if and only if there exists a solution of the Schröder functional equation $F \circ f = \gamma F$ which is holomorphic in $$\mathbb D$$ and satisfies the condition $\frac{zF'(z)}{F(z)} = \frac{p(0)}{p(z)}\,,$ where $$p$$ is holomorphic in $$\mathbb D$$, $$\text{Re}\,{p(z)}>0$$ for $$z \in \mathbb D$$ and $$e^{-p(0)}=\gamma$$.
Theorem 2. Let $$f \in \mathcal{P} \setminus \mathcal{I}$$. Then $$f$$ is embeddable if and only if there exists a solution of the Abel functional equation $F \circ f = F+1$ which is holomorphic in $$\mathbb D$$ and satisfies the condition $\text{Re}{[(1-z^2)F'(z)]}>0$ for $$z \in \mathbb D$$.

##### MSC:
 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 39B12 Iteration theory, iterative and composite equations 39B32 Functional equations for complex functions
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