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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 \(\mathcal{P}[0] \setminus \mathcal{I}\) and \(\mathcal{P}[1] \setminus \mathcal{I}\). Now, we can state two of the author’s results.
Theorem 1. Let \(f \in \mathcal{P}[0] \setminus \mathcal{I}\), 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 \(f \in \mathcal{P}[1] \setminus \mathcal{I}\). 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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