Circle homeomorphisms and continued fractions

For an orientation preserving homeomorphism $f: \mathbb{T} \longrightarrow \mathbb{T}$ of the circle $\mathbb{T}=\mathbb{R}/ \mathbb{Z}$ with an irrational rotation number $\alpha_{f}$, we build karyon tilings $\mathcal{T}^{l}$ of levels $l=0,1,2,\ldots$ that are quasi-invariant with respect to $f$ and have minimal kernels. These tilings are used to obtain approximations for the rotation number $\alpha_{f}$ by continued fractions.