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) = logamp on m = 3 + 21 period point of U -sequence RLR21, where amp is the largest real root of equation am-2am-2-1 = 0; hj(f) = (1/2)j log2 on the boundary between 2j-1 band and 2j 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 RLaRb period points, we also get the relation of hR*Q(f)=(l/2)hQ(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.