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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."… – Joseph O'Rourke Jan 29 '12 at 18:04
I realize this comment comes rather late given when you asked the question, but what on earth is $[|\mathcal{M}|]$? Do you just want to build the flag complex on the graph represented by the transition matrix? – Vidit Nanda Aug 3 '13 at 15:48
@Vidit Nanda: $|\mathcal{M}|$ is the cardinality of $\mathcal{M}$. $[n] := \{1,\dots,n\}$. This is standard (if perhaps not common) usage AFAIK. – Steve Huntsman Aug 3 '13 at 17:22
up vote 3 down vote accepted

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.

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 $3$!

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