On the fractal dimension of turbulent black holes

We present measurements of the fractal dimension of a turbulent asymptotically anti-deSitter black brane reconstructed from simulated boundary fluid data at the perfect fluid order using the fluid-gravity duality. We argue that the boundary fluid energy spectrum scaling as $E(k)\sim k^{-2}$ is a more natural setting for the fluid-gravity duality than the Kraichnan-Kolmogorov scaling of $E(k) \sim k^{-5/3}$, but we obtain fractal dimensions $D$ for spatial sections of the horizon $H\cap\Sigma$ in both cases: $D=2.584(1)$ and $D=2.645(4)$, respectively. These results are consistent with the upper bound of $D=3$, thereby resolving the tension with the recent claim in Adams, Chesler, Liu (2014) that $D=3+1/3$. We offer a critical examination of the calculation which led to their result, and show that their proposed definition of fractal dimension performs poorly as a fractal dimension estimator on $1$-dimensional curves with known fractal dimension. Finally, we describe how to define and in principle calculate the fractal dimension of spatial sections of the horizon $H\cap \Sigma$ in a covariant manner, and we speculate on assigning a `bootstrapped' value of fractal dimension to the entire horizon $H$ when it is in a statistically quasi-steady turbulent state.