The article "Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics" by William Parry and Mark Pollicott, published in *Astérisque* (1990), provides a comprehensive study of zeta functions and periodic orbits in the context of hyperbolic dynamics. The authors develop the theory from the basics of subshifts of finite type and function spaces to advanced topics such as the Ruelle operator, entropy, Gibbs measures, and pressure. They explore the relationship between the spectra of Ruelle operators and the poles of zeta functions, which is crucial for understanding the analytic properties of these functions. The main results of the article include theorems that describe the distribution of closed orbits in hyperbolic systems, both in terms of time, space, and symmetry. These theorems are derived using methods inspired by analytic number theory and involve the analysis of general zeta functions. Specifically, the authors present a first-order asymptotic for the number of closed orbits of least period, a uniform distribution result for closed orbits, and an analogue of the Chebotarev theorem for closed orbits. The article also covers the spectral properties of the complex Ruelle operator, the analyticity properties of zeta functions, and the description of the Sinai–Ruelle–Bowen measure. It includes detailed proofs and discussions on various topics, such as the reduction to the case of aperiodic matrices, the Perron-Frobenius theorem for matrices, and the variational principle. The authors conclude with five appendices that provide additional background material, including the Wiener-Ikehara Tauberian theorem, unitary cocycles, Markov partitions, geodesic flows, and perturbation theory for linear operators. The article aims to present a unified account of their joint and separate work since 1983, placing it in its proper context within the broader field of dynamical systems and ergodic theory.The article "Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics" by William Parry and Mark Pollicott, published in *Astérisque* (1990), provides a comprehensive study of zeta functions and periodic orbits in the context of hyperbolic dynamics. The authors develop the theory from the basics of subshifts of finite type and function spaces to advanced topics such as the Ruelle operator, entropy, Gibbs measures, and pressure. They explore the relationship between the spectra of Ruelle operators and the poles of zeta functions, which is crucial for understanding the analytic properties of these functions. The main results of the article include theorems that describe the distribution of closed orbits in hyperbolic systems, both in terms of time, space, and symmetry. These theorems are derived using methods inspired by analytic number theory and involve the analysis of general zeta functions. Specifically, the authors present a first-order asymptotic for the number of closed orbits of least period, a uniform distribution result for closed orbits, and an analogue of the Chebotarev theorem for closed orbits. The article also covers the spectral properties of the complex Ruelle operator, the analyticity properties of zeta functions, and the description of the Sinai–Ruelle–Bowen measure. It includes detailed proofs and discussions on various topics, such as the reduction to the case of aperiodic matrices, the Perron-Frobenius theorem for matrices, and the variational principle. The authors conclude with five appendices that provide additional background material, including the Wiener-Ikehara Tauberian theorem, unitary cocycles, Markov partitions, geodesic flows, and perturbation theory for linear operators. The article aims to present a unified account of their joint and separate work since 1983, placing it in its proper context within the broader field of dynamical systems and ergodic theory.