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Warning: My education in formal mathematics is very weak so I apologize for any confusions/errors in the following, please don't hesitate to correct me.

Question: Consider a smooth dynamical system $x_{t+1}=f(x_t)$ the trajectories of which describe a smooth manifold in the state space. I would like to measure distances (at least locally) between two points $y_i$ and $y_j$ on the manifold. Naively, I feel that $f$ has some information about how far any two points are. For example, let's say I have three points $y_1$, $y_2$ and $y_3$ on the manifold, then the number of iterations of $f$ it takes to go from $y_1$ to $y_2$ and $y_1$ to $y_3$ tells me something about the distance between $y_1$ and $y_2$ and between $y_1$ and $y_3$.

So, under what conditions does $f$ allow me to measure distances between any points on the manifold described by $x_{t+1}=f(x_t)$? And when it's possible to do this, is there an explicit relationship between $f$ and the metric it induces?

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    $\begingroup$ You tagged differential equations: do you actually mean $\dot{x} = f(x)$ for your dynamical system? If not, do you mean to tag using "difference equations"? For clarification: by the phrase "the trajectories of which describe a smooth manifold in the state space", do you mean that each trajectory describe a smooth manifold, or are you considering the collection of all trajectories as a smooth manifold? $\endgroup$
    – Willie Wong
    Feb 5, 2018 at 15:44
  • $\begingroup$ No. Consider a period-2 orbit $y_1\rightarrow y_2 \rightarrow y_1$. It maps points arbitrarily far away to each other in 1 iteration. $\endgroup$
    – Piyush Grover
    Feb 5, 2018 at 15:53
  • $\begingroup$ @WillieWong Thanks for the comment. Yes, I am particularly interested in differential equations but didn't want the question to sound too specific. I mean the collection describes a smooth manifold. So let's say a torus described by quasiperiodic motion. I am tempted to include strange attractor, but I am not sure about that. $\endgroup$
    – Sujaan
    Feb 6, 2018 at 3:26
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    $\begingroup$ Ok, let me try to reformulate what you wrote to see if I understand your question. You are given a dynamical system and you are interested in whether the system induces a sort of metric on an arbitrary (?) invariant submanifold of your system. Is that more-or-less correct? There's something I still don't understand: if you are just interested in an invariant manifold $M$, there is no guarantee that given any arbitrary $y, z\in M$ that they belong to the same orbit. What would the distance between them in this case? $\endgroup$
    – Willie Wong
    Feb 6, 2018 at 22:14

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