Iterated function: Difference between revisions

Revision as of 00:22, 27 October 2023

Revision as of 00:39, 27 October 2023

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When the points of the orbit converge to one or more limits, the set of [[accumulation point]]s of the orbit is known as the '''[[limit set]]''' or the '''ω-limit set'''.

The ideas of attraction and repulsion generalize similarly; one may categorize iterates into [[stable manifold|stable set]]s and [[unstable set]]s, according to the behaviourbehavior of small [[Neighbourhood (mathematics)|neighborhood]]s under iteration. (Also see [[Infinite compositions of analytic functions]].)

Other limiting behavioursbehaviors are possible; for example, [[wandering point]]s are points that move away, and never come back even close to where they started.

==Invariant measure==

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Note: these two special cases of {{math|''ax''² + ''bx'' + ''c''}} are the only cases that have a closed-form solution. Choosing ''b'' = 2 = –''a'' and ''b'' = 4 = –''a'', respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table.

Some of these examples are related among themselves by simple conjugacies.

Some of these examples are related among themselves by simple conjugacies. A few further examples, essentially amounting to simple conjugacies of Schröder's examples can be found in ref.<ref>{{Cite journal | last1 = Katsura | first1 = S. | last2 = Fukuda | first2 = W. | doi = 10.1016/0378-4371(85)90048-2 | title = Exactly solvable models showing chaotic behavior | journal = Physica A: Statistical Mechanics and Its Applications | volume = 130 | issue = 3 | pages = 597 | year = 1985 | bibcode = 1985PhyA..130..597K }}</ref>

==Means of study==

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* [[Functional square root]]

* [[Abel function]]

* [[Schröder's equation]]

* [[Böttcher's equation]]

* [[Infinite compositions of analytic functions]]

* [[Flow (mathematics)]]