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Persistent homology of Markovian dynamical systems

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.

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Marian Mrozek seems to be an expert in this general area. E.g., see the talk slides, "Computational homology in dynamical systems." hamilton.nuigalway.ie/DeBrunCentre/SecondWorkshop/… – Joseph O'Rourke Jan 29 2012 at 18:04