We prove a generalization of the Conley conjecture: Every Hamiltonian diffeomorphism of a closed symplectic manifold has infinitely many periodic orbits if the first Chern class vanishes on the second fundamental group. In particular, this removes the rationality condition from similar theorems by Ginzburg and Gürel. The proof in the irrational case involves several new ingredients including the definition and the properties of the filtered Floer homology for Hamiltonians on irrational manifolds. For this proof, we develop a method of localizing the filtered Floer homology for short action intervals using a direct sum decomposition. One of the summands only depends on the behavior of the Hamiltonian in a fixed open set and enables us to use tools from more restrictive cases in the proof of the Conley conjecture. We also prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians, which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits.

In this thesis, we study Reeb dynamics on prequantization circle bundles and the filtered (equivariant) symplectic homology of prequantization line bundles, aka negative line bundles, with symplectically aspherical base. We define (equivariant) symplectic capacities, obtain an upper bound on their growth, prove uniform instability of the filtered symplectic homology and touch upon the question of stable displacement. We also introduce a new algebraic structure on the positive (equivariant) symplectic homology capturing the free homotopy class of a closed Reeb orbit - the linking number filtration - and use it to give a new proof of the non-degenerate case of the contact Conley conjecture (i.e., the existence of infinitely many simple closed Reeb orbits), not relying on contact homology.

For a symplectic vector space, recall the identification of the symplectic group Sp(V) with an open and dense subset of the Lagrangian Grassmannian via the map sending each linear symplectomorphism to its graph. Our central result is in extending the mean index, using this embedding and a formal construction of the mean index in terms of a `circle map' on Sp(V), from the set of continuous paths in Sp(V) to those contained in a subset L with a complement of codimension two in the Lagrangian Grassmannian. Namely, we continuously extend the square of the circle map to a circle-valued map on L so that by applying the aforementioned construction to this new map, we reduce the existence and continuity of our extended index to the simpler problem of producing this continuous extension. Our secondary results concern the algebraic properties of the extended index with respect to a set-theoretic composition operation on linear relations which extends the usual group structure of Sp(V) to that of a monoid on the Lagrangian Grassmannian. To derive these we define an open and dense subset of differentiable paths in L which we call `stratum-regular', equip them with an equivalence relation and show that the point-wise composition of any two equivalent stratum-regular paths is piece-wise differentiable. We then show that over the stratum-regular paths, the extended index is homogeneous (for non-negative integers) and satisfies a quasimorphism-type bound for any equivalent pair of paths.