On the Dynamical System Generated by the Möbius transformation at Prime Times

Abstract

We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the Möbius map $x\mapsto (ax+b)/(cx+d)$ over a finite field of $p$ elements at the moments of time that correspond to prime numbers. In particular, we obtain nontrivial estimates of exponential sums with such sequences.