Universal Fine Structure of the Chaotic Region in Period-Doubling Systems

A new relation is reported which quantitatively describes the fine structure of the chaotic region of period-doubling systems. The relation determines the onset of fundamental periods and of ergodic behavior. It involves bifurcation rates ${\gamma }_k$, which converge to a new universal constant $\gamma =2.948\mathrm{}05\dots$. This theory is in agreement with numerical determinations of ${\gamma }_k$.