Infinite compositions of analytic functions

Frequently, continued fractions, series, products and other infinite expansions can be characterized as infinite compositions of analytic functions (ICAF), and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. It addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions.

Notation

There are several notations describing infinite compositions, including the following:

Forward compositions:

${{\text{F}}_{k,n}}{\text{(z)}}={\text{ }}{{\text{f}}_{\text{k}}}\circ {{f}_{k+1}}\circ \cdots \circ {{f}_{n-1}}\circ {{f}_{n}}(z)$

Backward compositions:

${{G}_{k,n}}(z)={{f}_{n}}\circ {{f}_{n-1}}\circ \cdots \circ {{f}_{k+1}}\circ {{f}_{k}}(z)$.

Convergence is interpreted as the existence of ${\underset {n\to \infty }{\mathop {\lim } }}\,{{\text{F}}_{1,n}}{\text{(z)}}$ and ${\underset {n\to \infty }{\mathop {\lim } }}\,{{G}_{1,n}}(z)$.

Contraction theorem

Most results can be considered extensions of the following contraction theorem for analytic functions:

Let $f$ be analytic in a simply-connected region $S$ and continuous on the closure ${\overline {S}}$ of $S$. Suppose $f\left({\overline {S}}\right)$ is a bounded set contained in $S$. Then ${{F}_{n}}(z)=f\circ f\circ \cdots \circ f(z)\to \alpha$, the attractive fixed point of $f$ in $S$ , for all $z\in {\overline {S}}$.

Infinite compositions of contractive functions

Forward (or inner or right) compositions:

Let $\left\{{{f}_{n}}\right\}$ be a sequence of functions analytic on a simply-connected domain $S$. Suppose there exists a compact set $\Omega \subset S$ such that for each n, ${{f}_{n}}(S)\subset \Omega$. Then $\left\{{{F}_{1,n}}(z)\right\}$ converges uniformly on $S$ to a constant function $F(z)=\lambda$.

Backward (or outer or left) compositions:

Let $\left\{{{f}_{n}}\right\}$ be a sequence of functions analytic on a simply-connected domain $S$. Suppose there exists a compact set $\Omega \subset S$ such that for each n, ${{f}_{n}}(S)\subset \Omega$. Then $\left\{{{G}_{1,n}}(z)\right\}$ converges uniformly on $S$ to $\alpha \in \Omega$ if and only if the sequence of fixed points $\left\{{{\alpha }_{n}}\right\}$ of the $\left\{{{f}_{n}}\right\}$ converge to $\alpha$.