Escape from the unit interval under the transformation $x\mapsto \lambda x(1-x)$
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- by Bruce R. Henry PDF
- Proc. Amer. Math. Soc. 41 (1973), 146-150 Request permission
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
- P. R. Stein and S. M. Ulam, Non-linear transformation studies on electronic computers, Rozprawy Mat. 39 (1964), 66. MR 169416
- 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 316636, DOI 10.1016/0097-3165(73)90033-2 S. J. McNaughton and B. R. Henry, A difference model of population growth: model properties and biological reality, 1971 (unpublished).
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 146-150
- MSC: Primary 26A18
- DOI: https://doi.org/10.1090/S0002-9939-1973-0322109-7
- MathSciNet review: 0322109