Just to clarify, your setting is, for Riemannian manifold; ergodic w.r.t. and for all the Lyapunov exponents is neigative?

If that's the right setting, then can the support even be a periodic orbit? (somehow I can only imaging the support being a fixed point)

Can we just do the following: take any two point , join them by a minimal geodesic, consider then length of the image of the geodesic as a curve from $f^n(p)$ to $f^n(q)$ by integration the length of the curve tends to as $n \rightarrow \infty$ so the distance between any two points is tending to with an contraction constant at least so diameter of the support should be .

I must have misunderstood something about your question…

]]>Nice blog~^^

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