COUNTING PROBLEM FOR SOME RANDOM CONFORMAL ITERATED FUNCTION SYSTEMS

The paper’s Theorem 4 correctly proves that for a.e. environment the random counting function has liminf 0 and limsup ∞ under the construction in Section 5, but its statement implicitly relies on exponential-length word counts N_n(ξ) ≍ r(A)^n. This fails for the irreducible 2×2 two-cycle A=[[0,1],[1,0]] where r(A)=1, in which case N_n(ξ)≡1 and the ratio reduces to N^ω_ξ(T)→∞ (so the liminf cannot be 0). The proof cites a deterministic inequality that presumes r(A)>1, yet the theorem’s hypotheses do not exclude r(A)=1. The candidate’s solution makes the missing hypothesis explicit and treats the r(A)=1 case separately, so it is correct and sharper.