External and internal rays in (super)attracting basins and corresponding puzzle pieces belong to the fundamental tools in polynomial dynamics. They have been used with great success by many researchers, including Douady and Hubbard and Yoccoz. The authors of this article introduce this terminology for rational functions with a certain type of parabolic dynamics. More precisely, for \(d>1\) and \(v_d:=\frac{d-1}{d+1}\) consider the Blaschke product \(P_d(z):= \frac{z^d + v_d}{1 + v_d z^d}\). The map \(P_d\) has the unit disk as a forward invariant parabolic basin of the double fixed point \(z=1\). In \(\mathbb{D}\), \(P_d\) has a single critical point at \(z=0\) of order \(d-1\). On the unit circle, \(P_d\) is conjugate to \(z\mapsto z^d\), and hence one can extend combinatorial concepts such as itineraries to the maps \(P_d\). This leads to a senseful concept of parabolic rays for these Blaschke products in analogy to rays for \(z\mapsto z^d\). Now, if \(f\) is a holomorphic map which has a fixed parabolic basin with a unique critical point of degree \(d-1\), then on this basin, the dynamics of \(f\) is conjugate to the dynamics of \(P_d\) on \(\mathbb{D}\). Hence one can carry over the introduced concepts to this more general setting.
For the special case of \(d=2\), Petersen and Roesch define parabolic puzzles using the idea of Yoccoz puzzle pieces for quadratic polynomials. While the introduction of rays is rather straightforward, the construction of parabolic puzzles requires a careful and extensive technical setup. As an application, the authors consider the set \(Per_1(1)\) of Möbius conjugacy classes of rational maps of degree \(2\) with parabolic fixed point of multiplier \(1\). They prove that the Julia set of any non-renormalizable map in the connectedness-locus of \(Per_1(1)\) is locally connected.