一维单峰映象的拓扑熵

在对Logistic映象数值计算的基础上,我们分析了一维单峰映象的逆轨道结构,证明了不同参数处逆轨道总数N(n)随求逆次数n而变化的递推公式。借此解析地求得了在倍周期区中h(f)≡0;在U序列RLR²¹的m=3+2l周期点上h(f)=logα_mp,其中α_mp为方程α^m-2α^m-2-1=0的最大实根;在2^j-1常和2j带交界处h_j(f)=(1/2)^jlog2,由此可得聚点μ_∞处拓扑熵的标度指数t=0.449806…。在此基础上,我们还求得了混沌区的周期窗口,U序列RL^aR^b所对应的各点处的拓扑熵,以及h_R*Q(f)=(1/2)h_Q(f)的关系。证明是在M.S.S.规则和“*”乘法则的基础上进行的。所以本文的结果对一维单峰映象是普适的。关键词：

THE TOPOLOGICAL ENTROPY OF ONE-DIMENSIONAL UNIMODAL MAPS

On the basis of computational calculation for the logistic map, we analyse the inverse orbital structure of one-dimensional unimodalmap and prove the different recurrence formulae which show that the total number of inverse orbit N(n) changes with inverting time n at different parameters. By this way, we analytically obtain h(f)=0 within the period-doubling region; h(f) = loga_mp on m = 3 + 21 period point of U -sequence RLR²¹, where a_mp is the largest real root of equation a^m-2a^m-2-1 = 0; h_j(f) = (1/2)^j log2 on the boundary between 2^j-1 band and 2^j band, from this result we find the scaling exponent of topological entropy near the accumulation point μ_∞, t = 0.449806…. By means of above conclusion, we obtain the topological entropy within each "window" in chaotic region and on U-sequence RL^aR^b period points, we also get the relation of h_R*Q(f)=(l/2)h_Q(f). Because we carry out our proofs on the basis of M.S.S. rule and "*" composition law, the results in this paper are universal to all the one-dimensional unimodal maps.