Exponential rate of decay of correlations of equilibrium states associated with non-uniformly expanding circle maps

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 $(\omega ,\mathrm{\Omega })$ may be a useful element for the development of equilibrium theory. Here we identify a particular feature of modulus $\mathrm{\Omega }$ (precisely ${\lim }_{x\to 0^{+}}{\sup }_{\mathsf{d}}\mathrm{\Omega }\left(\mathsf{d}x\right)\mathrm{/}\mathrm{\Omega }(\mathsf{d})=0$) 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 $\mathrm{\Omega }$ 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