  This work proposes a new approach for mapping GPU threads onto a family of
discrete embedded 2D fractals. A block-space map $\lambda:
\mathbb{Z}_{\mathbb{E}}^{2} \mapsto \mathbb{Z}_{\mathbb{F}}^{2}$ is proposed,
from Euclidean parallel space $\mathbb{E}$ to embedded fractal space
$\mathbb{F}$, that maps in $\mathcal{O}(\log_2 \log_2(n))$ time and uses no
more than $\mathcal{O}(n^\mathbb{H})$ threads with $\mathbb{H}$ being the
Hausdorff dimension of the fractal, making it parallel space efficient. When
compared to a bounding-box (BB) approach, $\lambda(\omega)$ offers a
sub-exponential improvement in parallel space and a monotonically increasing
speedup $n \ge n_0$. The Sierpinski gasket fractal is used as a particular case
study and the experimental performance results show that $\lambda(\omega)$
reaches up to $9\times$ of speedup over the bounding-box approach. A
tensor-core based implementation of $\lambda(\omega)$ is also proposed for
modern GPUs, providing up to $\sim40\%$ of extra performance. The results
obtained in this work show that doing efficient GPU thread mapping on fractal
domains can significantly improve the performance of several applications that
work with this type of geometry.
