Consider a dynamical system $(T,X)$ that admits a Markov partition $\mathcal{M}$ (e.g., an Anosov map), and consider the corresponding 0-1 transition matrix $A$. It is commonplace to study information about the growth of the number of closed orbits as a function of iterates through a zeta function.
However, it seems to me that there may be another approach based on persistent homology (which would be interesting). Assume for convenience that $X$ is connected, so that $A$ corresponds to a connected graph. Consider the usual graph distance and the filtration of Vietoris-Rips complexes on $[|\mathcal{M}|]$. Intuitively I would guess that the dimension 1 persistent homology would encode the same sort of information as the zeta function. But this approach could give higher dimensional information.
So my question is: using the graph distance for the transition matrix of a dynamical system, can we use the persistent homology of the corresponding Vietoris-Rips filtration to obtain useful information about the dynamical system?
Addendum: I realize that this might involve generalized persistence, as the partition diameter could come into play.
