Give a example about invariant ergodic measure and quasi-symmetric mapping

Question

Is there a example $(X,f,\mu)$ such that $X$ is a closed subset of Euclidean space, $f$ be a quasi-symmetric mapping but not a Lipschitz mapping, $f(X)=X$, $\mu$ is a finite measure on $X$ that is invariant and ergodic under $f$.

Answer 1

Take an irrational rotation $f:S^1\to S^1$, which is well-known to be ergodic. Of course it is also bi-Lipschitz. But we can conjugate it by a quasisymmetric non-bi-Lipschitz homeomorphism $\varphi:S^1\to S^1$ such as $\varphi(\exp({i\theta}))=\exp({i\theta|\theta|/\pi})$, $-\pi\le \theta\le \pi$. The homeomorphism $g=\varphi^{-1}\circ f\circ \varphi$ is quasisymmetric and ergodic. To check ergodicity, just notice that $g^{-1}(E)=E$ implies $f^{-1}(\varphi(E))=\varphi(E)$.