A criterion of vanishing of the spectral density based on homoclinic sums

Let (X, d) be a compact metric space, let T: X→X be a homeomorphism satisfying a certain suitable hyperbolicity assumption, and let μ be a Gibbs measure on X relative to T. Let λ be a complex number |λ|=1, and let f:X → ? be a Hölder continuous function. It is proved that $sumlimits_{k in mathbb{Z}} {lambda ^{ - k} } left( {intlimits_X {f(T^k x)bar f(x)mu (dx) - left| {intlimits_X {f(x)mu (dx)} } right|^2 } } right) = 0$ if and only if ∑λ^?k(f(T^ky) ? f(T^kx)) = 0 for all x, y ε X such that $d(T^k x,T^k y)xrightarrow[{|k| to infty }]{}0$ . Bibliography: 11 titles.