Lyapunov Exponents and Invariant Measures on a Projective Bundle

A discrete dynamical system generated by a diffeomorphism $f$ on a compact manifold is considered. The Morse spectrum is the limit set of Lyapunov exponents of periodic pseudotrajectories. It is proved that the Morse spectrum coincides with the set of averagings of the function $\varphi(x,e)=\ln|Df(x)e|$ over the invariant measures of the mapping induced by the differential $Df$ on the projective bundle.