Density of periodic measures and large deviation principle for generalised mod one transformations

We introduce a new class of piecewise monotonic maps, called generalised mod one transformations, which include all $(\alpha,\beta)$ and generalised $\beta$-transformations, and prove that all transitive generalised mod one transformations with positive topological entropy satisfy the level-2 large deviation principle with a unique measure of maximal entropy. This is obtained by our result on the density of periodic measures for more general class of piecewise mototonic maps, which is a partial answer to the open problem posed by Hofbauer and Raith (1998 Fund. Math.157 221–34); Raith (www.math.iupui.edu/~mmisiure/open/PR1.pdf).