Persistence of Morse decompositions over grid resolution for maps and time series

We can approximate a continuous self-map $f$ of a compact metric space by discretizing the space into a grid. Through either the map itself or a time series, $f$ induces a multivalued grid map $\mathcal F$. The dynamical properties of $\mathcal F$ depend on the resolution of the grid, and we study the persistence of these properties as we change the resolution. In particular, we look at the persistence of Morse decompositions, at both the global (Morse graph) and local (individual Morse set) levels, using several notions of persistence -- graph structure, persistent homology, and mixing properties.