Density of periodic measures and large deviation principle for generalized (α, β)-transformations

The paper’s Theorem A states precisely that every transitive generalized (α,β)-transformation satisfies the level‑2 LDP with respect to its unique measure of maximal entropy, and it proves this by (i) a general criterion (Proposition A) that transitive piecewise monotone maps with dense periodic measures satisfy the LDP, and (ii) showing density of periodic measures for generalized (α,β)-transformations (Theorem B). This matches the model’s plan of reducing to a general criterion and then verifying that criterion via Hofbauer’s Markov diagram, so the proofs are essentially the same. The LDP is defined exactly as in the model’s statement, with reference to the MME m (the paper recalls uniqueness of the MME under transitivity and positive entropy) . The paper’s Proposition A and Theorem B appear and are combined to yield Theorem A exactly as the model describes . The only extra assertion made by the model is an explicit identification of the rate function J(ν)=h_top(T)−h_ν(T) (for T‑invariant ν, +∞ otherwise); while consistent with standard large deviations frameworks, this explicit formula is not stated in the paper. Apart from that embellishment, the reasoning aligns with the paper’s argument.