Disjointness of the Möbius Transformation and Möbius Function

We study the distribution of the sequence of elements of the discrete dynamical system generated by the Möbius transformation x↦(ax+b)/(cx+d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \mapsto (ax + b)/(cx + d)$$\end{document} 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 Möbius function.