Explore the fourth installment of a five-part mini-course on symbolic dynamics for nonuniformly hyperbolic systems. Delve into advanced topics such as the Improvement Lemma, surface maps, invariant manifolds, and extensions within the framework of Markov Partitions and Young Towers in Dynamics. Learn how these concepts contribute to the study of nonuniformly hyperbolic systems and their statistical properties. Gain insights into the similarities and differences between Young Towers and infinite Markov Partitions approaches, as well as their applications in various dynamical systems. Enhance your understanding of these powerful techniques used to construct SRB measures and analyze complex hyperbolic systems with discontinuities or singularities.