MEASURES AND GENERALIZED BRATTELI DIAGRAMS FOR DYNAMICS OF INFINITE ALPHABET-SUBSTITUTIONS

The paper proves that for a bounded-size, left determined substitution on a countably infinite alphabet with irreducible, aperiodic, recurrent substitution matrix M, there exists a shift-invariant measure on X_σ and it is finite iff the Perron–Frobenius left eigenvector ℓ satisfies ∑ℓ_i<∞ (Theorem 7.6), by constructing a tail-invariant measure on a stationary generalized Bratteli diagram with cylinder masses ℓ_v/λ^n (Theorem 7.4) and transporting it via a Borel isomorphism (Theorem 6.13) . The candidate solution reaches the same conclusion using a stationary Bratteli diagram together with a suspension by the roof function ρ(a)=|σ(a)|, then pushing forward to X_σ; this is a standard but different route than the paper’s direct isomorphism without suspension. Both arguments agree on the measure formula and the finiteness criterion; the model’s extra suspension step is not used in the paper but is consistent with it.