  In this paper we use the theory of barcodes as a new tool for studying
dynamics of area-preserving homeomorphisms. We will show that the barcode of a
Hamiltonian diffeomorphism of a surface depends continuously on the
diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms.
  Our main dynamical application concerns the notion of {\it weak conjugacy},
an equivalence relation which arises naturally in connection to $C^0$
continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for
a large class of Hamiltonian homeomorphisms with a finite number of fixed
points, the number of fixed points, counted with multiplicity, is a weak
conjugacy invariant. The proof relies, in addition to the theory of barcodes,
on techniques from surface dynamics such as Le Calvez's theory of transverse
foliations.
  In our exposition of barcodes and persistence modules, we present a proof of
the Isometry Theorem which incorporates Barannikov's theory of simple Morse
complexes.
