Expanding interval maps with intermittent behaviour, physical measures and time scales

We exhibit a new family of piecewise monotonic expanding interval maps with interesting intermittent-like statistical behaviours. Among these maps, there are uniformly expanding ones for which a Lebesgue-typical orbit spends most of the time close to an "indifferent Cantor set" which plays the role of the usual neutral fixed point. There are also examples with an indifferent fixed point and an infinite absolutely continuous invariant measure. Like in the classical case, the Dirac mass at $0$ describes the statistical behaviour at usual time scale while the infinite one tells about the statistical behaviour at larger scales. But, here, there is another invariant measure describing the statistical behaviour of the ergodic sums at a third natural (intermediate) time scale.
To try to understand this last phenomenon, we propose a more general construction that yields an example for which we conjecture there is an infinite number of natural time scales.