For the map T, reduced fractions a/M of fixed denominator M have finite order under iteration with natural density one, described by residue classes of a modulo M^{n+1} for exact order n.
Moreira,https://mathoverflow.net/questions/377706/ an-alternative-to-continued-fraction-and-applications#comment958452_ 377706 10
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Rational dynamics of a prime-representing map
For the map T, reduced fractions a/M of fixed denominator M have finite order under iteration with natural density one, described by residue classes of a modulo M^{n+1} for exact order n.