Some Exact Results in the Functional Map f(z;μ,z)= 1-μ|χ|^z

Chyi-Lung Lin

《Chinese Journal of Physics》 33卷2期 (1995/04) Pp. 117-132

並列摘要

We show that in the functional map f(z;μ,z)= 1-μ|χ|^z, where 0 ≤ μ ≤ 2 and z ≥ 1, there exists a convergent sequence {∑n(z)}n=2^∞ formed from the parameter values of μ. For each n, C,(z) is the largest parameter value among those μn(z)'s for which χ = 0 is the period-n point of the map f. Each C,(z) signifies that in the range ∑n(z) ≤ μ ≤ 2 there can be only one parameter value , μn+1(z) for which χ = 0 is the period-(n + 1) p oint off. The only , μn+1(z) in the range C,(z) ≤ μ ≤ 2 is ∑n+1(z). These C,(z)'s satisfy the ordered relation C,(z) ＜ ∑n+1(z) ＜ 2, and form a convergent sequence called the marginal supercycle parameter value sequence. The limit and the asymptotic convergence ratio of this convergent sequence can be exactly calculated. The results are lim(n→∞) C,(z) = 2 and 6(z) ≡ lim(n→∞) ∑(z)- ∑n-1(z)/∑n+1(z)-∑n(z) = 2z. Considering the graph (μ, f^n+1(0; μ, z)) in the range ∑(z) ≤ μ ≤ 2 and the graph (μ, f^n”(0; u, z)) in the range ∑n-1(z) ≤ μ ≤ 2, we find that there is the asymptotic scaling behavior: fn+1(0; μ,z) = f^n(0;2 - k(2 - μ),z), where the scaling factor k = 2z. We expect that there is also scaling behavior in the neighborhood of χ = 0 : f^n+1(χ; ∑n+1(χ), z)) = f^n(sχ;∑n(z),z)), with the scaling factor s = (2z)^1/z. These results show that the iteration constants are z-dependent.