Large deviation principle for linear mod 1 transformations

For $0\le \alpha <1$ and $\beta >2$, we consider a linear mod 1 transformation on a unit interval; $x\mapsto \beta x+\alpha$ ($\mathrm{m}\mathrm{o}\mathrm{d}1$), and prove that it satisfies the level-2 large deviation principle with the unique measure of maximal entropy. For the proof, we use the density of periodic measures and Hofbauer’s Markov Diagram.