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

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.