The quasi-shadowing 性质 for nonuniformly partially hyperbolic systems

In this paper, we establish a new quasi-shadowing 性质 for any nonuniformly partially hyperbolic set of a $C^{1+\alpha}$ diffeomorphism, which is adaptive to the movement of the pseudo-orbit. Moreover, the quasi-specification 性质 and the quasi-closing 性质 are also investigated. As an application of the quasi-closing 性质, we extend Katok's result on the growth of periodic orbits for the hyperbolic ergodic measure to any ergodic measure: the number of quasi-periodic points grows exponentially in at least the metric entropy.