The high-emergence conjecture for differentiable dynamics

For a differentiable dynamical system, its metric emergence at scale $\epsilon>0$ is the minimal number of probability measures needed to approximate the empirical-measure statistics within precision $\epsilon$. The system has high emergence when

$$\limsup_{\epsilon\to 0}\frac{\log \mathscr E_{\mathrm{Leb}}(\epsilon)}{-\log\epsilon}=\infty.$$

High-emergence conjecture. Dynamics of high emergence are typical in many senses and contexts. This is stated as the first aim of the authors' program on emergence; the source leaves the meanings of “typical” and “many senses and contexts” intentionally broad.