Renormalization of binary trees derived from one-dimensional unimodal maps

Abstract

For one-dimensional unimodal mapsh _λ(x):I →I, whereI=[x ₀,x ₁] when λ=λ_max, a binary tree which includes all the periodic windows in the chaotic regime is constructed. By associating each element in the tree with the superstable parameter value of the corresponding periodic interval, we define a different unimodal map. After applying a certain renormalization procedure to this new unimodal map, we find the period-doubling fixed point and the scaling constant. The period-doubling fixed point depends on the details of the maph _λ(x), whereas the scaling constant equals the derivative$h'_{\lambda _{max} } \left( {x_0 } \right)$. The thermodynamics and the scaling function of the resulting dynamical system are also discussed. In addition, the total measure of the periodic windows is calculated with results in basic agreement with those obtained previously by Farmer. Up to 13 levels of the tree have been included, and the convergence of the partial sums of the measure is shown explicitly. A new scaling law has been observed, i.e., the product of the length of a periodic interval characterized by sequenceQ and the scaling constant ofQ is found to be approximately 1.