具抛物不动点的二维保测度映射及其三维扩张的KS熵

KS ENTROPY OF MEASURE-PRESERVING MAPPINGS WITH PARABOLIC FIXED POINT

我们已经研究了分别具椭圆和双曲不动点的二维保测度映射及其受摄三维扩张的ＫＳ熵。本文研究一类具抛物不动点的二维保测度映射：及其受摄扩张：的ＫＳ熵随参数Ａ、Ｂ、Ｃ、Ｄ、Ｅ的变化．数值探索结果表明：适当定义区域内的二维映射Ｔ２的ＫＳ熵与Ａ无关，与我们的理论分析结果相一致。受摄扩张映射Ｔ３的ＫＳ熵随摄动参数Ｂ、Ｃ、Ｄ的增大而增大，却随Ｅ的增大而减小．我们还发现，随着摄动的逐渐增强，映射Ｔ３的不变环面将逐渐破裂，使更多的轨道逃逸，从而可能使映射Ｔ３的ＫＳ熵减小。另外，不变环面存在的判别式在大范围内仍在一定程度上有效。

We have studied the KS entropy of a 2-dimensional measure-preserving mapping with an elliptic or hyperbolic fixed point respectively, as well as that of its 3-dimensional extension. In this paper we discuss the variations of the KS entropy of a 2-dimensional measure-preserving mapping with a parabolic fixed point:and its perturbed extension where A,B,C,D,E are parameters.Numerical exploration shows that: the KS entropy of the 2-dimensional mapping T2 in a suitably defined region in the phase space is independent of the parameter A, which is in concord with the theoritical analyses. As for the perturbed extension T3, the KS entropy increases with the parameters B,C,D but decreases with the parameter E. However, with the enhancement of perturbations, the invariant tori of the mapping T3 will gradually breakup, leading to the increase of escape orbits and possibly decrease of the KS entropy. Moreover, we find that the criterion obtained in [3], for the existence of the invariant tori,seems to be still available in large scale.