Monodromy through bifurcation locus of the Mandelbrot set

We investigate the behavior of itinerary sequence of each point of the Julia set of $z\mapsto z^2 + c$ when the parameter $c$ in the shift locus is allowed to pass through points in the bifurcation locus $\mathcal{P}_2$, which we call ``narrow", first proposed by Dierk Schleicher in \cite{schleicher2017internal}. We first show the combinatoric and geometric properties of narrow characteristic arcs. Also, we show how the itinerary sequence changes in an algorithmic way by using lamination models proposed by Keller in \cite{keller2007invariant}. Finally, we found an equivalence relation on the set of $0$-$1$ sequences so that the changing rule is a shift invariant up to the equivalence relation. This generalizes Atela's works in \cite{atela1992bifurcations}, \cite{atela1993mandelbrot}, which dealt with the special case of the generalized rabbit polynomials.