This book, "Symplectic Invariants and Hamiltonian Dynamics" by Helmut Hofer and Eduard Zehnder, explores the theory of symplectic geometry and Hamiltonian systems. It presents a systematic treatment of symplectic invariants, particularly symplectic capacities, and their applications to Hamiltonian dynamics. The authors discuss the rigidity of symplectic mappings, the existence of periodic orbits, and the relationship between symplectic and volume-preserving mappings. The book is based on lectures given at several institutions and includes detailed proofs of key results. The first chapter introduces symplectic manifolds, symplectic diffeomorphisms, and Hamiltonian vector fields. It also discusses the classification of 2-dimensional symplectic manifolds and the concept of the Birkhoff normal form. The second chapter introduces symplectic capacities, which are invariants that measure the size of symplectic embeddings and are used to study rigidity phenomena in symplectic geometry. The third chapter provides a detailed construction of a specific symplectic capacity, $ c_0 $, which is defined using Hamiltonian systems and measures the minimal $ C^0 $-oscillation of a Hamiltonian function. The fourth chapter applies the capacity $ c_0 $ to the question of whether a compact energy surface carries a periodic orbit. It demonstrates that many global existence results can be derived from this invariant. The chapter also discusses the Weinstein conjecture and its solution by C. Viterbo. The fifth chapter studies the subgroup of symplectic diffeomorphisms of $ \mathbb{R}^{2n} $ generated by time-dependent Hamiltonian vector fields. It introduces the Hofer metric, a bi-invariant metric on this subgroup, and discusses its relationship with the symplectic capacity $ c_0 $ and the displacement energy. The sixth chapter focuses on the fixed point theory for Hamiltonian mappings on compact symplectic manifolds. It presents a proof of the Arnold conjecture, which states that a Hamiltonian mapping has at least as many fixed points as a real-valued function on the manifold has critical points. The proof uses the action principle and relates the problem to Floer homology and symplectic homology. The book concludes with an appendix that provides technical details and references to topological tools used in the text.This book, "Symplectic Invariants and Hamiltonian Dynamics" by Helmut Hofer and Eduard Zehnder, explores the theory of symplectic geometry and Hamiltonian systems. It presents a systematic treatment of symplectic invariants, particularly symplectic capacities, and their applications to Hamiltonian dynamics. The authors discuss the rigidity of symplectic mappings, the existence of periodic orbits, and the relationship between symplectic and volume-preserving mappings. The book is based on lectures given at several institutions and includes detailed proofs of key results. The first chapter introduces symplectic manifolds, symplectic diffeomorphisms, and Hamiltonian vector fields. It also discusses the classification of 2-dimensional symplectic manifolds and the concept of the Birkhoff normal form. The second chapter introduces symplectic capacities, which are invariants that measure the size of symplectic embeddings and are used to study rigidity phenomena in symplectic geometry. The third chapter provides a detailed construction of a specific symplectic capacity, $ c_0 $, which is defined using Hamiltonian systems and measures the minimal $ C^0 $-oscillation of a Hamiltonian function. The fourth chapter applies the capacity $ c_0 $ to the question of whether a compact energy surface carries a periodic orbit. It demonstrates that many global existence results can be derived from this invariant. The chapter also discusses the Weinstein conjecture and its solution by C. Viterbo. The fifth chapter studies the subgroup of symplectic diffeomorphisms of $ \mathbb{R}^{2n} $ generated by time-dependent Hamiltonian vector fields. It introduces the Hofer metric, a bi-invariant metric on this subgroup, and discusses its relationship with the symplectic capacity $ c_0 $ and the displacement energy. The sixth chapter focuses on the fixed point theory for Hamiltonian mappings on compact symplectic manifolds. It presents a proof of the Arnold conjecture, which states that a Hamiltonian mapping has at least as many fixed points as a real-valued function on the manifold has critical points. The proof uses the action principle and relates the problem to Floer homology and symplectic homology. The book concludes with an appendix that provides technical details and references to topological tools used in the text.