Editing Dyadic transformation (section)

==Relation to tent map and logistic map==
The dyadic transformation is [[topologically semi-conjugate]] to the unit-height [[tent map]]. Recall that the unit-height tent map is given by

:<math>x_{n+1} = f_1(x_n) = \begin{cases}
    x_n   & \mathrm{for}~~ x_n \le 1/2 \\ 
    1-x_n & \mathrm{for}~~ x_n \ge 1/2 
\end{cases}</math>

The conjugacy is explicitly given by
:<math>S(x)=\sin \pi x</math> 
so that

:<math>f_1 = S^{-1} \circ T \circ S</math>

That is, <math>f_1(x) = S^{-1}(T(S(x))).</math> This is stable under iteration, as
:<math>f_1^n = f_1\circ\cdots\circ f_1 = S^{-1} \circ T \circ S \circ S^{-1} \circ \cdots \circ T \circ S = S^{-1} \circ T^n \circ S</math>

It is also conjugate to the chaotic ''r''&nbsp;=&nbsp;4 case of the [[logistic map]].  
The ''r''&nbsp;=&nbsp;4 case of the logistic map is <math>z_{n+1}=4z_n(1-z_n)</math>; this is related to the [[bit shift]] map in variable ''x'' by

:<math>z_n =\sin^2 (2 \pi x_n).</math>

There is also a semi-conjugacy between the dyadic transformation (here named angle doubling map) and the [[Complex quadratic polynomial|quadratic polynomial]]. Here, the map doubles angles measured in [[turn (angle)|turns]]. That is, the map is given by 
:<math>\theta\mapsto 2\theta\bmod 2\pi.</math>