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- In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system. Although the title of this article specifies analytic functions, there are results for more general functions of a complex variable as well. (en)
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- {Gn} converges uniformly on compact subsets of S to γ ∈ Ω if and only if the sequence of fixed points {γn} of the {fn} converges to γ. (en)
- If fn → f where f is parabolic with fixed point γ. Let the fixed-points of the {fn} be {γn} and {βn}. If
then Fn → λ, a constant in the extended complex plane, for all z. (en)
- If an ≡ 1,
then Fn → F is entire. (en)
- Suppose where there exist such that and implies and Furthermore, suppose and Then for (en)
- Set ε'n = suppose there exists non-negative δ'n, M1, M2, R such that the following holds:
Then Gn → G is analytic for < R. Convergence is uniform on compact subsets of {z : < R}. (en)
- Let f be analytic in a simply-connected region S and continuous on the closure of S. Suppose f is a bounded set contained in S. Then for all z in there exists an attractive fixed point α of f in S such that: (en)
- {Fn} converges uniformly on compact subsets of S to a constant function F = λ. (en)
- Suppose is a simply connected compact subset of and let be a family of functions that satisfies
Define:
Then uniformly on If is the unique fixed point of then uniformly on if and only if . (en)
- On the set of convergence of a sequence {Fn} of non-singular LFTs, the limit function is either:
a non-singular LFT,
a function taking on two distinct values, or
a constant.
In , the sequence converges everywhere in the extended plane. In , the sequence converges either everywhere, and to the same value everywhere except at one point, or it converges at only two points. Case can occur with every possible set of convergence. (en)
- Suppose where there exist such that implies Furthermore, suppose and Then for (en)
- Let φ be analytic in S = {z : < R} for all t in [0, 1] and continuous in t. Set
If ≤ r < R for ζ ∈ S and t ∈ [0, 1], then
has a unique solution, α in S, with (en)
- If {Fn} converges to an LFT, then fn converge to the identity function f = z. (en)
- If fn → f and all functions are hyperbolic or loxodromic Möbius transformations, then Fn → λ, a constant, for all , where {βn} are the repulsive fixed points of the {fn}. (en)
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- Theorem (en)
- Backward Compositions Theorem (en)
- Contraction Theorem for Analytic Functions (en)
- Forward Compositions Theorem (en)
- Theorem E1 (en)
- Theorem E2 (en)
- Theorem FP2 (en)
- Theorem GF3 (en)
- Theorem GF4 (en)
- Theorem LFT1 (en)
- Theorem LFT2 (en)
- Theorem LFT3 (en)
- Theorem LFT4 (en)
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- In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system. (en)
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- Infinite compositions of analytic functions (en)
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