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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).

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    $\begingroup$ Never seen the map (I think), but let me note what one can say right away: it is topologically conjugate to $2x\bmod1$ and it has an absolutely continuous invariant measure (due to the clear control of first and second derivatives). The only nontrivial matter is to find the conjugacy explicitly, then everything follows. $\endgroup$ – John B Feb 1 '18 at 17:13
  • $\begingroup$ Thanks John B. Interestingly, the function that conjugates T(x) to the doubling map 2x mod1 is the Minkowski's question mark function ?(x) $\endgroup$ – Luke Feb 2 '18 at 20:54
  • $\begingroup$ I have seen the variation where the second case is just $(1-x)/x$ so that it is continuous. In that form it is conjugate to the tent map, to $4x(1-x)$, etc and the orbit itinerary of $x$ counts out the continued fraction for $x$ in some way that is easy to see with a couple of examples, but I don't remember right now. $\endgroup$ – Ned Feb 2 '18 at 22:19
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    $\begingroup$ You may - I dare say, you should - reply your own question. It may help visibility for future searchers $\endgroup$ – Rolazaro Azeveires Feb 2 '18 at 23:09
  • $\begingroup$ Indeed. Please do not answer questions (even your own) in an edit. Post a self answer instead. $\endgroup$ – Wrzlprmft Feb 3 '18 at 8:22
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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.

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