"Dynamical" spectral gap for the orignal system out of the spectral gap for the induced system I would like to prove presence of a spectral gap for the transfer (Ruelle-Perron-Frobenius) operator for some non-uniformly hyperbolic dynamical system on the unit interval. Suppose that I know how to prove presence of a spectral gap for the transfer operator of the induced system.
Is there some standard approach (and a good reference for it) that would allow to get presence of a spectral gap for the original system? 
As far as I understand usually inducing is used together with coupling-like approach and the spectrum of the transfer operator does not appear explicitly.
There are some papers where people work with spectral gaps for transfer operators for induced systems but those that I have seen are focused on more delicate problems (linear response, for example) and it is not that easy to extract the essence of the argument from them.
 A: 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. 
A: There are two standard methods for pulling back results about decay of corellations from an induced system: Young towers, and the operator renewal theory introduced by Sarig and developed by Gouëzel. I don't know that either can be used to deduce actual spectral information, but perhaps the latter is slightly more likely.
An entirely different strategy could be to work with the transfer operator on a Hardy space, where the size of the derivative is unimportant and spectral gaps instead come from the way in which the backward contractions affect the boundary of the domain. (I use this trick in my paper on the Binary Euclidean algorithm.) If your map is analytic then this could be an option.
