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Let $X$ be a compact metric space. Let $f:X\rightarrow X$ be a contraction map. I need to find all $f$-invariant Borel probability measures.

Thank you.

Alp Uzman
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adtx11
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    Intuitively, since it is a compact metric space,successive applications of f (on a set B=X) will tend to the unique fixed point. Since you want an invariant measure, it has to be supported on that fixed point. – nonlinearism Feb 15 '13 at 16:38
  • See http://mathoverflow.net/questions/121874/finding-invariant-borel-probability-measures-for-a-contraction-map – UwF Feb 15 '13 at 17:42

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Hint: Suppose $\mu$ is $f$-invariant. Let $B_0 = X$ and $B_{n+1} = f(B_n)$. Then $\mu(B_n) = 1$ for all $n$. What can you say about the diameter of $B_n$? What does this imply about $B := \bigcap_n B_n$?

Nate Eldredge
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