  We study the distribution of the sequence of elements of the discrete
dynamical system generated by the M\"obius transformation $x \mapsto (ax +
b)/(cx + d)$ over a finite field of $p$ elements. Motivated by a recent
conjecture of P. Sarnak, we obtain nontrivial estimates of exponential sums
with such sequences that imply that trajectories of this dynamical system are
disjoined with the M\"obius function.
