  Julia and Mandelbrot sets, which characterize bounded orbits in dynamical
systems over the complex numbers, are classic examples of fractal sets. We
investigate the analogs of these sets for dynamical systems over the hyperbolic
numbers. Hyperbolic numbers, which have the form $x+\tau y$ for $x,y \in
\mathbb{R}$, and $\tau^2 = 1$ but $\tau \neq \pm 1$, are the natural number
system in which to encode geometric properties of the Minkowski space
$\mathbb{R}^{1,1}$. We show that the hyperbolic analog of the Mandelbrot set
parameterizes connectedness of hyperbolic Julia sets. We give a
wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia
sets.
