How random is a chaotic system? The study of statistical properties of dynamical systems has been a central theme in dynamics. The breakthrough of D. Dolgopyat in his study of decay of correlations in Anosov flows revolutionized this field and the related field of Prime Orbit/Geodesic Theorems. The main idea is to analyze fine properties of the Ruelle operator and the dynamical zeta functions. Such results have been established in many dynamical systems thanks to the works of W. Parry, M. Pollicott, V. Baladi, D. Dolgopyat, C. Liverani, L. Stoyanov, G. A. Margulis, A. Avila, S. Gouezel, J. C. Yoccoz, M. Tsujii, and many others.

In this talk, we are going to introduce a brief history of such results, focusing mainly on the works of F. Naud, H. Oh, and D. Winter on hyperbolic rational maps. We are going to discuss the main ideas used to obtain such results. If time permits, we are going to discuss how to extend such results to a class of non-hyperbolic rational maps known as (rational) expanding Thurston maps. This is a work-in-progress joint with T. Zheng.