Reducible expansions and related sharp crossovers in Feigenbaum’s renormalization field

We discuss reducible aspects of Mao and Hu’s multiple scaling expansion [ J. Stat. Phys. 46, 111 (1987); Int. J. Mod. Phys. B 2, 65 (1988)] in the framework of renormalization theory. After establishing a suitable form of reduced expansion, we present numerical evidence showing sharp crossovers from Feigenbaum's constant ( δ ) to Mao and Hu’s constant ( δ ′ ) in the first-order reduced expansion. We find that the crossover is caused by the universal scaling relation existing in constant coefficients of Mao and Hu’s expansion. Special attention is paid to constant coefficients corresponding to scaling terms including δ ′ . We show numerically that they converge to zero in universal ways with convergence ratios larger than δ . Here, the convergence direction is transversal to the unstable eigendirection of the linearized renormalization operator. From this observation, we propose a concise form of expansion for Feigenbaum’s universal function g r ( x ) .