We study the distribution of the sequence of elements of the discrete dynamical system generated by the Möbius 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 Möbius function.

中文翻译：

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莫比乌斯变换与莫比乌斯函数的不相交性

我们研究了由Möbius变换\（x \ mapsto（ax + b）/（cx + d）\）生成的离散动力学系统的元素序列在p个元素的有限域上的分布。受P. Sarnak的最新猜想的启发，我们获得了具有此类序列的指数和的非平凡估计，这意味着该动力学系统的轨迹与Möbius函数不相交。