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.

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. She studies the Pommeville-Manneau maps and proves polynomial decay of correlations. On the other hand, if you induce on the expanding interval, the map has derivative everywhere bigger than 1. This induced map has exponential decay of correlations.