Multifractal analysis for Markov interval maps with countably many branches

The paper states and proves three main results (Theorems 2.1, 2.2, 2.4) giving conditional variational formulas for the Hausdorff dimension of Birkhoff-average level sets in the recurrent part (general case), under strong expansion, and under C0-type hypotheses, respectively. The definitions of Z, Z0, α1, α2, α3, s∞ and α4, and the statements of these theorems match the candidate’s targets exactly. The paper’s proofs hinge on bounded distortion, a careful upper-bound covering via cylinders (using product/Bernoulli constructions and Abramov’s formula on a finite-entropy suspension), approximation of measures by ergodic measures on finite subshifts, and entropy-at-infinity δ∞ to handle escape; see the statements for Theorems 2.1, 2.2 and 2.4 and the proof infrastructure in Sections 5–8 . The candidate’s solution reproduces the same conclusions with a close but not identical route: periodic-loop approximations for Z, measure-dimension h/λ heuristics, a covering argument akin to Katok-type estimates, and a two-scale gluing/covering using δ∞ and L in the C0 case. This is broadly consistent with the paper’s approach and conclusions. One notable overstatement in the candidate write-up is the blanket claim in the C0 setting that the transient set Λ_T has dimension δ∞/L. The paper does not assert this equality in general; instead it proves dim Λ(0) = max{α4(0), dim Λ_T} and interprets the zero-measure contribution in α4(0) as δ∞/L (without identifying dim Λ_T with δ∞/L in full generality) . Aside from this minor overshoot, the candidate’s outline aligns with the paper’s correct results.