Characterization of the natural measure by unstable periodic orbits in chaotic attractors

The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of the unstable periodic orbits embedded in that set. This result has been proven to be valid for hyperbolic chaotic systems. We test the goodness of such a periodic-orbit characterization of the natural measure for nonhyperbolic chaotic systems by comparing the natural measure of a typical chaotic trajectory with that computed from unstable periodic orbits. Our results suggest that the unstable periodic-orbit formulation of the natural measure is typically Valid for nonhyperbolic chaotic systems.