Spatial ergodicity dimension: Ergodic self-similarity of trajectories in discrete chaotic systems

Chaotic systems exhibit self-similarity as a manifestation of their inherent complexity. In this paper, it is found that the spatial ergodic trajectories of chaotic systems exhibit consistent traversal behaviors across different spatial scales, manifesting power-law scaling: $\beta \left(\varepsilon \right)\propto {\varepsilon }^{-d}$ with $d>1$. Here, $d$ is introduced as a novel metric of fractal dimensions termed the Spatial Ergodicity Dimension ($D_E$), which provides a quantitative assessment of the statistical self-similarity in the spatial ergodic trajectories of discrete chaotic systems and fills the research gap in qualitative ergodicity analysis. $D_E$ reflects the uniformity of the spatial motion state distribution of chaotic systems over a finite time period, exhibiting scale invariance. Experimental results demonstrate that the $D_E$ varies across chaotic systems, unveiling that a lower $D_E$ corresponds to more even spatial traversal trajectories. Furthermore, the $D_E$ is closely associated with the invariant probability measure of chaotic systems. Specifically, chaotic systems with different invariant probability measures exhibit distinct $D_E$ values, while different chaotic systems with the same invariant probability measure exhibit similar $D_E$ values.