  We study the period doubling renormalization operator for dynamics which
present two coupled laminar regimes with two weakly expanding fixed points. We
focus our analysis on the potential point of view, meaning we want to solve
$$V=\mathcal{R} (V):=V\circ f\circ h+V \circ h,$$ where $f$ and $h$ are
naturally defined. Under certain hypothesis we show the existence of a explicit
``attracting'' fixed point $V^*$ for $\mathcal{R} $. We call $\mathcal{R}$ the
renormalization operator which acts on potentials $V$. The log of the
derivative of the main branch of the Manneville-Pomeau map appears as a special
``attracting'' fixed point for the local doubling period renormalization
operator. We also consider an analogous definition for the one-sided 2-full
shift $\S$ (and also for the two-sided shift) and we obtain a similar result.
Then, we consider global properties and we prove two rigidity results. Up to
some weak assumptions, we get the uniqueness for the renormalization operator
in the shift. In the last section we show (via a certain continuous fraction
expansion) a natural relation of the two settings: shift acting on the
Bernoulli space $\{0,1\}^\mathbb{N}$ and Manneville-Pomeau-like map acting on
an interval.
