Math 2.0

Sequences, series, summability Definition

Infinite compositions of functions

Builds on Function compositionMetric spaceLimit of a sequence

Definition

Let DD be a subset of a metric space, for example of ℂ\C, and let f1,f2,…f_1, f_2, \dots be functions from DD to DD. The forward (or inner) compositions are Fn=f1∘f2∘⋯∘fnF_n = f_1 \circ f_2 \circ \cdots \circ f_n. The backward (or outer) compositions are Gn=fn∘fn−1∘⋯∘f1G_n = f_n \circ f_{n-1} \circ \cdots \circ f_1. The infinite forward composition converges at a point z∈Dz \in D if the sequence (Fn(z))(F_n(z)) has a limit, and similarly for the backward composition. When the limit exists for all zz in DD, it defines the function F(z)=lim⁡n→∞Fn(z)F(z) = \lim_{n \to \infty} F_n(z) (or G(z)=lim⁡n→∞Gn(z)G(z) = \lim_{n \to \infty} G_n(z)).

Remarks

Example: on D=ℝD = \R let fn(x)=x2+1f_n(x) = \frac{x}{2} + 1 for all nn. Then Gn(0)=2−21−nG_n(0) = 2 - 2^{1-n}, so the backward composition converges at 00 to the fixed point 22 of f1f_1. Non-example: with fn(x)=−xf_n(x) = -x for all nn, the sequence Fn(1)=(−1)nF_n(1) = (-1)^n has no limit, so the infinite composition does not converge at 11. For a continued fraction with fn(w)=anbn+wf_n(w) = \frac{a_n}{b_n + w}, the values Fn(0)F_n(0) are the convergents; backward compositions do not have this interpretation. When all fnf_n equal a single map ff, both FnF_n and GnG_n are the iterate f∘nf^{\circ n}.

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Element, definition, no evidence levelElementdefinitionMetric space, concept, no evidence levelMetric spaceconceptProposition, concept, no evidence levelPropositionconceptSequence, definition, no evidence levelSequencedefinitionSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionFunction composition, definition, no evidence levelFunction compositiondefinitionInfinite compositions of functions, definition, no evidence levelInfinite compositions…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Metric space concept
  3. Proposition concept
  4. Sequence definition
  5. Set definition
  6. Implication definition
  7. Predicate definition
  8. Real sequence definition
  9. Ordered pair definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Limit of a sequence definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Function composition definition
  19. Infinite compositions of functions definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Infinite_compositions_of_analytic_functions
About this page
TagY8RZ73N0JA
URIm2://Y8RZ73N0JA
Kinddefinition
WritingW0 stub
MSC40A30, 30D05
DefinesInfinite compositions of functions
UsesFunction composition, Metric space, Limit of a sequence
Changed2 November 2026

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