Interplay between Floer homology and Hamiltonian dynamic

In this thesis we primarily focus on the interplay between Floer homology and Hamiltonian dynamics. This has been an active area of research since the late 1980s with the introduction of pseudo-holomorphic curves by Gromov and Floer homology by Floer. The Floer Homology has been a very powerful tool to study dynamics on a symplectic or contact manifold and the subject is very broad. Here primarily we concentrate on three aspects of the connections of Floer Homology and dynamics. Firstly, the connection between Hamiltonian dynamics and symplectic topology of the underlying manifold by studying special kind of Hamiltonians such as toric pseudo-rotations. We further study two Floer-theoretic invariants of symplectic and contact dynamics: ``barcode entropy'' and symplectic capacities. We use these invariants to understand various Hamiltonian dynamics behaviors such as pseudo-rotations (Hamiltonians with finitely many orbits) or other extremes (Hamiltonians with infinitely many orbits or very chaotic dynamics)