Homoclinic sums criterion for vanishing of spectral density

Let $(X,d)$ be a compact metric space, $T\colon X\to X$ be a homeomorphism satisfying certain suitable hyperbolicity hypothesis and $\mu$ be a Gibbs measure on $X$ relative to $T$. The following statement is proved in the paper.
Let $\lambda$ be a complex number with $|\lambda|=1$ and $f\colon X\to\mathbb C$ be a Hölder continuous function. Then the equality
$$ \sum_{k\in\mathbb Z}\lambda^{-k}\Biggl(\int_Xf(T^kx)\overline f(x)\mu(dx)-\Bigg|\int_Xf(x)\mu(dx)\Bigg|^2\Biggr)=0 $$
holds true if and only if the identity
$$ \sum_{k\in\mathbb Z}\lambda^{-k}(f(T^ky)-f(T^kx))=0 $$
is valid for each $x,y\in X$ with the property that $d(T^kx,T^ky)\xrightarrow[|k|\to\infty]{}0$. Bibl. 11 titles.