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Solutions of Doubly and Higher Order Iterated Equations

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Abstract

Michael Fisher once studied the solution of the equation f(f(x))=x 2−2. We offer solutions to the general equation f(f(x))=h(x) in the form f(x)=g(ag −1(x)) where a is in general a complex number. This leads to solving duplication formulas for g(x). For the case h(x)=x 2−2, the solution is readily found, while the h(x)=x 2+2 case is challenging. The solution to these types of equations can be related to differential equations.

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REFERENCES

  1. Fisher, The iterated equation g(g(x))=x 2-2, private communication.

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Authors and Affiliations

  1. Naval Research Laboratory, 4555 Overlook Avenue, SW Washington, District of Columbia, 20375

    Becky A. Brown

  2. 4650 N. Washington Blvd., Arlington, Virginia, 22201

    A. Ray Brown

  3. Office of Naval Research, 800 Quincy Avenue Arlington, Virginia, 22210

    Michael F. Shlesinger

Authors
  1. Becky A. Brown
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  2. A. Ray Brown
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  3. Michael F. Shlesinger
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Brown, B.A., Brown, A.R. & Shlesinger, M.F. Solutions of Doubly and Higher Order Iterated Equations. Journal of Statistical Physics 110, 1087–1097 (2003). https://doi.org/10.1023/A:1022144826671

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  • DOI: https://doi.org/10.1023/A:1022144826671

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