  In the context of expanding maps of the circle with an indifferent fixed
point, understanding the joint behavior of dynamics and pairs of moduli of
continuity $ (\omega, \Omega) $ may be a useful element for the development of
equilibrium theory.
  Here we identify a particular feature of modulus $ \Omega $ (precisely $
\lim_{x \to 0^+} \sup_{\mathsf d} \Omega\big({\mathsf d} x \big) /
\Omega(\mathsf d) = 0 $)
  as a sufficient condition for the system to exhibit exponential decay of
correlations with respect to the unique equilibrium state associated with a
potential having $ \omega $ as modulus of continuity.
  This result is derived from obtaining the spectral gap property for the
transfer operator acting on the space of observables with $ \Omega $ as modulus
of continuity, a property that, as is well known,
  also ensures the Central Limit Theorem. Examples of application of our
results include the Manneville-Pomeau family
