We report on a recent joint work with Roman Muchnik.
Let M be a compact negatively curved manifold, G be its fundamental
group and X its universal cover. Denote the boundary of X by B. B is
endowed with the Paterson-Sullivan measure. The associated unitary
representation $L^2(B)$ is called a boundary unitary representation.
Fixing G, but changing the metric on M, we get a different boundary
(given by a different measure on the same topological boundary), and a
different boundary representation.
We will explain the setting and indicate the proof of
Theorem 1: The boundary representations are irreducible.
Theorem 2: Two boundary representations are equivalent if and only if
the associated marked length spectrums are the same (up to a scalar
multiple).
The marked length spectrum is the assignment associate to a free
homotopy class of closed loops in M the length of a shortest geodesic
in it.
The proof of the theorem is based on the mixing property of the
geodesic flow on M.