A generic-dimensional property of the invariant measures for circle diffeomorphisms (Q2437967)

Let \(\alpha\in {\mathbb R}/{\mathbb Z}\) be a Liouville number, i.e., \(\alpha\) is irrational and for any \(N\in {\mathbb N}\), there is \(p/q\in {\mathbb Q}\), \((p,q)=1\), such that \(|\alpha-p/q|< 1/q^N\). Denote by \(F_\alpha\) the set of all orientation-preserving \(C^\infty\) diffeomorphisms of the circle having the rotation number \(\alpha\). Note that any \(f\in F_\alpha\) has the unique invariant probability measure \(\mu_f\) which is related with the regularity of the conjugacy of \(f\) to \(R_\alpha\), the rigid rotation. \textit{V. Sadovskaya} [Ergodic Theory Dyn. Syst. 29, No. 6, 1979--1992 (2009; Zbl 1186.37032)] has shown that, for any \(d\in [0,1]\), \[ S_\alpha^d=\{ f\in F_\alpha \mid \text{dim}_H(\mu_f) =d\} \neq \emptyset. \] Here \(\text{dim}_H(\mu)\) is the Hausdorff dimension of a measure \(\mu\). In this paper, the author goes further and proves that the set \(S_\alpha^0\) contains a countable intersection of \(C^0\) dense and \(C^\infty\) dense subsets of \(F_\alpha\). This implies that generic diffeomorphisms in \(S_\alpha^0\) are not smoothly conjugate to \(R_\alpha\). The proof is based on the fast approximation with delicate estimates.