For the logistic map http://mathworld.wolfram.com/LogisticMapR=2.html the formula (4) in the link is valid only for $x₀<1/2$. How I can find a similar expression for $x₀>1/2$. The same question for $r=-2$ (defined only in an interval of lenght $0.5$). See here http://mathworld.wolfram.com/LogisticMapR=-2.html
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The same formula works after just one step, since $1/2-a$ and $1/2+a$ map to the same value. – Mark McClure Oct 18 '14 at 17:51
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@MarkMcClure: Can you please elaborate. – DER Oct 18 '14 at 17:52
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Since the expression $rx(1-x)$ is symmetric under $x\to1-x$, it's OK here to use absolute value signs in the logarithm:
$$x_n={1\over2}\left(1-\exp(2^n\ln\left|1-2x_0\right|)\right)$$
This now works for all $x_0$. (One caveat: when $n=0$, the expression on the right hand side returns the smaller of $x_0$ and $1-x_0$.)
Barry Cipra
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@ Barry Cipra: The same question for $r=-2$. See here http://mathworld.wolfram.com/LogisticMapR=-2.html. defined only in an interval of lenght $0.5$. – DER Oct 18 '14 at 18:27