Rational dynamics of a prime-representing map

We study the rational dynamics of the map $\mathcal{T}(x)=\lfloor x\rfloor(1+\{x\})$, which appears in the recursive construction of the prime-representing constant of Fridman, Garbulsky, Glecer, Grime and Florentin. For a rational number $x\geq 2$ with denominator $M$, we define its order to be the least non-negative integer $n$ such that $\mathcal{T}^n(x)$ is an integer, if such an $n$ exists, and ask whether every rational number has finite order. For each $n$, we prove that the reduced fractions $a/M$ of exact order $n$ are described by residue classes of $a$ modulo $M^{n+1}$, and give a recurrence for the number $A(n,M)$ of residue classes of exact order $n$. We then show that for each fixed denominator the fractions of finite order have natural density one among all reduced fractions with that denominator, which implies in particular that there is no infinite arithmetic progression of rational numbers of infinite order. We also give an explicit family of fractions of prescribed order for every denominator, and fully characterize the case $M=2$.