Abstract: Hamiltonian Floer homology, developed by Andreas Floer in the 1980s, has become one of the most powerful tools in symplectic geometry for studying Hamiltonian systems. In this talk, we study the topological entropy of Hamiltonian systems through persistences in Hamiltonian Floer homology.

In the first part of the talk, we establish a result known as braid stability on symplectic surfaces, for the spectral distance. This leads to two persistence-type results for topological entropy: effective robustness and lower semicontinuity. In particular, effective robustness means that the topological entropy does not drop by more than epsilon under perturbations supported in a disk whose area is bounded by a constant depending on epsilon.

In the second part, we define a Floer-theoretic entropy for Reeb flows via a variant of Hamiltonian Floer homology, referred to as symplectic homology barcode entropy. We show that the value of this entropy is independent of the choice of Liouville filling used in the definition. Furthermore, we prove that this barcode entropy provides a lower bound for the topological entropy.