Let
X
be a proper, geodesically complete Hadamard space, and
Γ<Is(X) a discrete subgroup
of isometries of
X
with the fixed point of a rank one isometry of
X
in its infinite limit set. In this paper we prove that if
Γ has
nonarithmetic length spectrum, then the Ricks–Bowen–Margulis measure
— which generalizes the well-known Bowen–Margulis measure in the
CAT(−1) setting — is
mixing. If in addition the Ricks–Bowen–Margulis measure is finite, then we also have equidistribution
of
Γ-orbit
points in
X,
which in particular yields an asymptotic estimate for the orbit counting function of
Γ.
This generalizes well-known facts for nonelementary discrete isometry
groups of Hadamard manifolds with pinched negative curvature and proper
CAT(−1)-spaces.
Keywords
rank one space, Bowen–Margulis measure, mixing,
equidistribution, orbit counting function
