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][1] on $[|\mathcal{M}|]$. Intuitively I would guess that the dimension 1 [persistent homology][2] 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. 

  [1]: http://en.wikipedia.org/wiki/Vietoris%25E2%2580%2593Rips_complex
  [2]: http://en.wikipedia.org/wiki/Topological_data_analysis#Persistent_homology