TUBULAR DIMENSION: LEAF-WISE ASYMPTOTIC LOCAL PRODUCT STRUCTURE, AND ENTROPY AND VOLUME GROWTH

The paper defines tubes and tubular dimension (Definitions 4.1 and 4.5) and proves Theorem 4.6: for μ-a.e. x, ρ^T_{i+1}(x)=Δ·(h_{i+1}−h_i)/χ_{i+1}+β_{i+1}χ_i d_i, using the intermediate-entropy formula for pucks (Theorem 3.3), the inclusion of pre-tubes into tubes (Proposition 4.4), and tubular covering/differentiation (Lemma 4.2) . The candidate solution independently obtains the same exponent by disintegrating μ along ξ_i inside ξ_{i+1}, combining the exact-dimensionality of leafwise conditionals and the transverse quotient (Ledrappier–Young), and leveraging Pesin–Lipschitz charts; this yields the same decay rate Δ·(h_{i+1}−h_i)/χ_{i+1}+β_{i+1}χ_i d_i. The paper’s proof avoids a direct product-disintegration estimate in favor of a covering/differentiation argument via pucks, but both are logically consistent and reach the same result. Ledrappier–Young’s relations di+1−di=(h_{i+1}−h_i)/χ_{i+1} underlie both approaches .