  We establish the existence and fundamental properties of the equilibrium
measure in uniformly quasiregular dynamics. We show that a uniformly
quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian
manifold admits an equilibrium measure $\mu_f$, which is balanced and invariant
under $f$ and non-atomic, and whose support agrees with the Julia set of $f$.
Furthermore we show that $f$ is strongly mixing with respect to the measure
$\mu_f$. We also characterize the measure $\mu_f$ using an approximation
property by iterated pullbacks of points under $f$ up to a set of exceptional
initial points of Hausdorff dimension at most $n-1$. These dynamical mixing and
approximation results are reminiscent of the Mattila-Rickman equidistribution
theorem for quasiregular mappings. Our methods are based on the existence of an
invariant measurable conformal structure due to Iwaniec and Martin and the
$\cA$-harmonic potential theory.
