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.