On the Dynamical System Generated by the M\"obius transformation at Prime Times

TL;DR

This paper investigates the distribution of sequences generated by Möbius transformations over finite fields at prime times, providing estimates for exponential sums and insights into their dynamical behavior.

Contribution

It introduces new bounds for exponential sums of Möbius dynamical sequences at prime iterations over finite fields.

Findings

Derived nontrivial estimates for exponential sums

Analyzed distribution properties of Möbius dynamical sequences

Provided insights into the behavior at prime iteration times

Abstract

We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the M\"obius 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.