On the Dynamical System Generated by the Möbius Transformation at Smooth Times

We study the distribution of the sequence of the first $N$ elements of the discrete dynamical system generated by the Möbius transformation $x\mapsto (\alpha x+\beta )\mathrm{/}(\gamma x+\delta )$ over a finite field of $p$ elements at the moments of time that correspond to $Q$-smooth numbers, that is, to numbers composed out of primes up to $Q$. In particular, we obtain nontrivial estimates of exponential sums with such sequences.