Density of periodic measures and large deviation principle for generalized $(\alpha ,\beta )$-transformations

We introduce generalized $(\alpha ,\beta )$-transformations, which include all $(\alpha ,\beta )$ and generalized $\beta$-transformations, and prove that all transitive generalized $(\alpha ,\beta )$-transformations satisfy the level-2 large deviation principle with a unique measure of maximal entropy. A crucial step in our proof is to establish density of periodic measures in the set of ergodic measures.