Problem 155

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(Thurston-Sullivan?) Are all smooth actions of $\Phi _g$ on $\mathbb S^1$ which are topologically conjugate to a standard one differentiably conjugate to the standard one?



Yes. [1]


  1. [shub1985expanding] Shub M, Sullivan D.  1985.  Expanding endomorphisms of the circle revisited. Ergodic Theory and Dynamical Systems. 5:285–289.

assuming we are talking about nielsens action of the mapping class group on the circle at infinity...the only question that I recall makes sense is to show this [ continuous even quasisymmetric ] action is not  conjugate to a smooth action [even C1 or even absolutely continuous] because the  absolute continuity is the obstruction to mostow rigidity in this dimension [ absolute continuity holding for quasiconformal or quasi symmetric  in higher dim.yielding mostow rigidity]




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