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](https://en.wikipedia.org/wiki/Vietoris%25E2%2580%2593Rips_complex) on $[|\mathcal{M}|]$. Intuitively I would guess that the dimension 1 [persistent homology](https://en.wikipedia.org/wiki/Persistent_homology) would encode the same sort of information as the zeta function. But this approach could give higher dimensional information.



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> 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?
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Addendum: I realize that this might involve generalized persistence, as the partition diameter could come into play. 



Persistent homology of Markovian dynamical systems
The answer to the question as stated seems to be "no".


Consider a three-element Markov partition $\mathcal{M} = \{A, B, C\}$ with directed edges $(A,B)$, $(B,C)$ and $(C,A)$. There is an obvious periodic orbit $A \to B \to C \to A$ but the Vietoris-Rips complex never has non-trivial persistent homology in dimension $1$ and completely fails to detect this orbit.


In particular, the Vietoris-Rips complex $\text{VR}\_t$ has three vertices for $t \in [0,1)$ and becomes a $2$-simplex for $t \geq 1$. One can generalize this construction to length $k$ periodic orbits to get a huge family of dynamical systems with non-trivial behavior but with contractible Vietoris-Rips complexes. The problem is that the Vietoris-Rips complex instantly "fills-up" all simplices whose $1$-skeleta are present.




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But if one uses something different from the Vietoris-Rips complex, then there is hope to extract topological invariants of your dynamics. Consider for example the following filtration: each vertex of the Markov graph is born at scale $0$, each edge is born according to the graph distance as usual, so up to the $1$-skeleton we agree with the Vietoris-Rips complex. But now, introduce each $d$-simplex (whose edges are all present) at the *sum* of all edge birth involved in its $1$-skeleton.


So for instance in the example above, we would have a length $2$ barcode in the $1$-dimensional persistent homology which captures the periodic orbit involving $A, B$ and $C$: the cycle forms at scale $1$ but the $2$-simplex does not make it a boundary until scale $3$! 


