TITLE: Is this chaotic map known?
QUESTION [5 upvotes]: Have you ever encountered the following map before?
$$x_{n+1}=T(x_n), x\in[0,1]$$ where
$$T(x)= \begin{cases} 
      \frac{x}{1-x} & x\leq 1/2 \\
      1-\frac{1-x}{x} & x> 1/2
   \end{cases}
$$
A quick exploration suggests that this map is chaotic but in the event this map is well-studied I could take advantage of what's already known.
Many thanks!
[EDIT] I found that this map is known as the modified Farey map. In case anyone else is interested, the following reference mentions it: 
C. Bonanno and S. Isola, Orderings of the rationals and dynamical systems, Colloquium Mathematicum 116, 2 (2009).

REPLY [3 votes]: I found that this map is known as the modified Farey map. In case anyone else is interested, the following reference mentions it: C. Bonanno and S. Isola, Orderings of the rationals and dynamical systems, Colloquium Mathematicum 116, 2 (2009).
In particular, the Minkowski's question mark function ?(x) conjugates this map to the shift map.