【作者】 张鹏飞；

【导师】 叶向东； 黄文； 夏志宏；

【作者基本信息】 中国科学技术大学 ， 基础数学， 2011， 博士

【摘要】 我们主要研究的是微分动力系统中的一致双曲以外的一些动力系统的性质。我们试图去理解那些在一致双曲系统中熟知的、非常理想的结论是否在在更大的一类系统中成立。我们的研究分成三个层次:部分双曲系统、控制分解系统和一般的微分自映射。首先我们证明,对一般的、具有正体积的部分双曲子集,该集合一定包含了‘相当多的’点上面的整体强稳定流形和整体强不稳定流形。如果一个正体积的部分双曲子集中有相当多的回复点,我们证明这个集合一定包含了一个正体积的、bi–saturated的子集。我们利用这些性质来深入研究accessible部分双曲系统的性质。如果一个accessible部分双曲系统有一个绝对连续不变概率测度(简称为ACIP),那么这个系统一定是传递的,该ACIP是全支撑的,并且关于该测度几乎所有的点的轨道都在整个流形上稠密。在center bunching的条件下,我们证明这样的系统至多存在一个ACIP,并且如果存在的话,这个测度其实是个光滑的测度:该测度相对于流形体积的Radon–Nikodym导数是Ho¨lder连续的,并且有上界和正的下界。其次我们考虑一个有整体控制分解的微分同胚。我们证明这样的系统一定有非平凡的真子系统,从而一定不是极小系统。这个证明主要用到了Man?e′的一个找出非回复点的手法和廖山涛先生关于控制分解的筛滤引理和跟踪引理。最后我们考虑最一般的一个可微自映射(一般是不可逆的)。这时候我们主要考虑该系统的一个有紧支撑的遍历测度的分形维数。我们证明该测度的下逐点维数不少于该测度的测度熵与该测度的最大的Lyapunov指数的比值。进一步,如果我们假设映射在该测度的支撑上非退化,并且最小的Lyapunov指数为正的,那么该测度的上逐点维数不超过该测度的测度熵与该测度的最小的Lyapunov指数的比值。再利用Young的一个经典的判别法则,我们给出了许多经典的维数型指标的类似的估计。最后我们证明,如果系统有多出的一点正则性(具体地说,即C1+α),那么前面非退化的要求可以去掉,即:如果最小的Lyapunov指数为正的,那么该测度的上逐点维数不超过该测度的测度熵与该测度的最小的Lyapunov指数的比值。对共形测度我们可以得到相当完备的刻画:如果一个C1+α的可微自映射的一个紧支撑的共形测度有正的Lyapunov指数,那么这个测度一定是恰当维数的,并且其分形维数恰为该测度的测度熵与其Lyapunov指数的比值。更多还原

【Abstract】 behaved uniformly hyperbolic systems. We try to catch the properties which mightpersist if we relax the uniformly hyperbolic assumption. We will go further step bystep: from partially hyperbolic systems to systems that only admit some dominatedsplitting, and then to general differentiable dynamics.Our first result concerns the dynamics of partially hyperbolic subsets/systems. Weshow that if a partially hyperbolic set is of positive volume, then it must contain‘many’global strong stable and unstable manifolds through it. We will show that a partiallyhyperbolic set has a bi–saturated subset of positive volume if the set of recurrent pointsis of positive volume. Then we carry on to describe the interesting dynamical proper-ties of partially hyperbolic systems. We show that if a partially hyperbolic system isessentially accessible and admits some ACIP, then the system is transitive, the ACIP issupported on the whole manifold, and almost every point with respect to the ACIP hasa dense orbit on the manifold. Moreover if the map is accessible and center bunched,then it admits at most one ACIP, and the ACIP, if exists, must be a smooth measure: theRadon–Nikodym derivative with respect to the volume is Ho¨lder continuous, boundedand bounded away from zero.Then we relax the restriction to a global dominated splitting. We show that ifa diffeomorphism admits a global dominated splitting, then it can not be a minimalsystem: there does exist some proper invariant subsystem. The proof mainly uses anargument due to Man?e′to locate some nonrecurrent point and Liao’s sifting lemma andshadowing lemma.Finally we will study the fractal dimensions of ergodic measures with compactsupport for general differentiable maps (not necessarily invertible). We show that thelower pointwise dimension of the ergodic measure is bounded from below by the ratioof the metric entropy and the largest Lyapunov exponent of that measure. Moreover ifthe map is non-degenerate on the support of the measure and the smallest Lyapunov ex- ponent is positive, we show that the upper pointwise dimension of the ergodic measureis bounded from above by the ratio of the metric entropy and the smallest Lyapunov ex-ponent of that measure. We give similar estimates for several classic characteristics ofdimensional type according to Young’s criterion. We can remove the non-degeneracycondition if the map has some extra regularity: assuming the map is C1+α, if the small-est Lyapunov exponent of the ergodic measure is positive, then the upper pointwisedimension of the measure is bounded from above by the ratio of the metric entropy andthe smallest Lyapunov exponent of that measure. We apply our results to the confor-mal ergodic measures: if the ergodic measure is conformal and has positive Lyapunovexponent, then it is exact dimensional with fractal dimension equal to the ratio of themetric entropy and the Lyapunov exponent.更多还原