There probably is not a spectral gap for your non-uniformly hyperbolic system. 

Maps with indifferent fixed points have polynomial decay of correlations (that is $a(n):=\int f\circ T^n g\,d\mu-\int f\,d\mu\,\int g\,d\mu$ decays polynomially). A spectral gap implies exponential decay of correlations.

See for example the 1999 paper of Lai-Sang Young: Recurrence times and rates of mixing.