Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms

The paper proves the claimed result (continuous arcs of ergodic measures with fixed center exponent χ and entropy ranging from 0 up to h_top(f, L(χ)) − ε) under BM1 hypotheses via a cascade of scu-horseshoes cyclically related to a blender and Feldman–Katok convergence (Theorem A and Theorem 8.1) , with a clear technical pipeline outlined in the Techniques section . By contrast, the candidate solution hinges on an unsubstantiated flip–flop scheme that asserts the existence of a hyperbolic basic set K_χ(ℓ) contained in the Lyapunov level set L(χ) with entropy arbitrarily close to h_top(f, L(χ)); this is not only unproved but also incompatible with χ = 0 (a hyperbolic basic set cannot have zero center exponent). It also assumes free concatenation and exponentially many exactly φ-balanced blocks without the careful ‘tails’ and return-time control the paper develops; these are nontrivial and are handled in the paper through cyclic relations and subordinated horseshoes, not by the asserted exact per-block balancing .