Equilibrium States for the Random β-Transformation through g-Measures

The candidate solution reproduces the paper’s construction and arguments: it builds the Markov partition and mixing SFT (Y,σ) for β∈B, defines the summable-variation potentials ϕ_{θ,α}, computes the explicit eigenfunction H_{θ,α} and the resulting g-function g_{θ,α}(y)=e^{θ(e0(y))}/∑_i e^{θ(i)}, shows µ(Y′)=1 and pushes forward to Kβ-invariant measures, and establishes non-productness for infinitely many θ. These steps match the paper’s proofs (Definition of B and coding; Lemma 4.2; Proposition 4.5; g-function identity; Theorem 5.1; Section 5.1) . Two minor discrepancies: (i) the paper states g_{θ,α}=g_{θ′,α′} iff θ=θ′, whereas equality holds modulo an additive constant to θ (the candidate notes this normalization-invariance); (ii) in part (d) the candidate invokes an unjustified shortcut for the 1/β relation that the paper derives rigorously via the Markov model. With that fix, the proofs coincide.