Has this chaotic map been studied?
This is the "Boole map", described by George Boole (yes, that George Boole) in his 1857 "On the Comparison of Transcendents, with Certain Applications to the Theory of Definite Integrals". A standard modern reference is the 1973 paper by R. Adler and B. Weiss, "The ergodic measure preserving transformation of Boole". It shows up in the answer to this MSE question.