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Proceedings of the American Mathematical Society

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Escape from the unit interval under the transformation $ x\mapsto\lambda x(1-x)$


Author: Bruce R. Henry
Journal: Proc. Amer. Math. Soc. 41 (1973), 146-150
MSC: Primary 26A18
MathSciNet review: 0322109
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Abstract: For $ \lambda > 4$ the transformation $ x \mapsto \lambda x(1 - x)$ maps the unit interval into itself except for an interval of $ x$ values centered at $ \tfrac{1}{2}$ that ``escapes". It is shown that almost all of the unit interval eventually escapes if the transformation is iterated. An easy example is then given for which the theorem fails, and the question is raised for exactly what class of functions the theorem holds.


References [Enhancements On Off] (What's this?)

  • [1] P. R. Stein and S. M. Ulam, Non-linear transformation studies on electronic computers, Rozprawy Mat. 39 (1964), 66. MR 0169416
  • [2] N. Metropolis, M. L. Stein, and P. R. Stein, On finite limit sets for transformations on the unit interval, J. Combinatorial Theory Ser. A 15 (1973), 25–44. MR 0316636
  • [3] S. J. McNaughton and B. R. Henry, A difference model of population growth: model properties and biological reality, 1971 (unpublished).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0322109-7
Keywords: One dimensional quadratic transformation, difference equation, infinite product, iteration, iterated inverse, oscillatory iteration
Article copyright: © Copyright 1973 American Mathematical Society