BEYOND 0 AND ∞: A SOLUTION TO THE BARGE ENTROPY CONJECTURE

The uploaded paper proves Theorem 1.1: for every r in [0,∞] there exists a pseudo-arc homeomorphism Hr with htop(Hr)=r, via an inverse-limit-of-pseudo-arcs construction and an adaptation of the Denjoy–Rees technique; the proof implants a strictly ergodic Bernoulli model with entropy r and uses variational principles to conclude htop=r . It also explains why earlier inverse-limit/natural-extension approaches cannot yield finite nonzero entropy on the pseudo-arc (results of Block–Keesling–Uspenskij and Mouron) . By contrast, the model’s solution asserts an almost one-to-one principal extension from a β-shift to a pseudo-arc obtained as an inverse limit of arcs, but provides no construction or justification; this runs counter to the obstacles summarized in the paper and does not match the paper’s method .