From Uniform Hyperbolicity To Differentiable Self-maps

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.