Generic dimensional and dynamical properties of invariant measures of full-shift systems over countable alphabets

The paper’s Theorem 1.2 proves exactly the three genericity statements the model claims—(1) PD = {μ : dimP(μ) = ∞} is dense Gδ, (2) {μ : R(x) = ∞ μ-a.e.} is dense Gδ, and (3) {μ : R(x,y) = ∞ (μ×μ)-a.e.} is residual—under the same hypotheses (full shift over any infinite Polish alphabet, with a product-compatible metric making the shift Lipschitz) . The proof structure in the paper matches the model’s outline: density via entropy-dense ergodic measures and a Lipschitz inequality translating entropy to lower packing dimension (Lemma 2.2) and then to recurrence (Proposition 2.3) ; Gδ-structure from prior work of the same authors (cited in the paper as [5]) and standard semicontinuity tools; and the waiting-time bound R(x,y) ≥ d̄μ(y) (Galatolo) to pass from infinite (upper) local dimension to infinite two-point indicator on μ×μ-a.e. pairs , together with the identification of packing dimensions with essential inf/sup of local dimensions (Proposition 1.1) . Minor differences are present only in how the Gδ-structure is justified (the model cites joint continuity of radius-packing functionals from earlier work; the paper cites Proposition 2.1 in [5]), but the arguments are substantially the same.