CONTRACTING ON AVERAGE ITERATED FUNCTION SYSTEMS BY METRIC CHANGE

The paper proves both parts rigorously: (i) under NEA on a compact space, LECA+ESCA imply the existence of n and λ with (F,p,dn,λ) contractive on average, and LECA⇔SA (Theorem 1.1), via a finite cover argument on K×K and Proposition 3.10 for the weighted metric dn,λ ; (ii) on S1 for C1 diffeomorphisms, proximality and “no common invariant measure” yield an α and k with (F,p,dα) k-ECA, hence D=(dα)k,λ is CA and d≤D≤C dα (Theorem 1.2) . The model’s Part A aligns in spirit and is essentially correct (it reproduces the k-ECA-to-CA upgrade for D and the LECA⇔SA equivalence). However, its Part B hinges on an unsubstantiated step: it claims Malicet’s theorem yields a uniform global bound E[dα(fn(x),fn(y))]≤Cρn dα(x,y) for all x,y,n, which is stronger than what is assumed or proved in the paper and is not justified under the paper’s hypotheses. The paper instead derives k-ECA for dα by a different route and then upgrades to CA via Proposition 3.10 .