On the rate of mixing of circle extensions of Anosov maps

Let $A: \mathbb T^2 \to \mathbb T^2$ be an Anosov diffeomorphism. Circle extensions $\widehat A$ are a rich family of non-uniformly hyperbolic diffeeomorphisms living on $\mathbb T^2 \times S^1$ for which the rate of mixing is conjectured to be generically exponential. In this paper, using transfer operators on Anisotropic Hilbert spaces, we investigate the possible rates of exponential mixing by exhibiting some explicit lower bounds on the decay rate by spectral techniques. The rates obtained are related to the topological pressure of two times the unstable jacobian.