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We prove an abstract result establishing that one can obtain the convergenceof Rare Events Point Processes counting the number of orbital visits to asequence of shrinking target sets from the convergence of corresponding pointprocesses for some induced system and matching shadowing shrinking sets insidethe base of the inducing scheme. We apply this result to prove a dichotomy fortwo classes of non-uniformly hyperbolic interval maps: Misiurewicz quadraticmaps and doubly intermittent maps. The dichotomy holds in the sense that theshrinking target sets may accumulate in any individual point $\zeta$ chosen inthe phase space and then one either obtains a limiting homogeneous Poissonprocess at every non-periodic point $\zeta$ or a limiting compound Poissonprocess with geometric multiplicity distribution at every periodic point. Wealso highlight the reconstruction performed in order to recover themultiplicity distribution for a periodic orbit sitting outside the base of theinduced map.

Inducing techniques for quantitative recurrence and applications to Misiurewicz maps and doubly intermittent maps