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Section 1 Function gallery

Subsection 1.1 Prototypical examples of bidegree \((1, n)\) RISPs

Consider the function \(\Phi\) given by

\begin{equation*} \Phi(z,w) = \left(-\frac{2 w z-w-z}{-w-z+2}, w\right). \end{equation*}

This function and basic variants are the foundational example in the study of rational inner functions in two variables.

(a) Vertical lines before mapping
(b) \(\Phi^n\) for \(n=1\)
(c) \(\Phi^n\) for \(n=2\)
(d) \(\Phi^n\) for \(n=3\)
Figure 1.2. Iteration of \(\Phi\)

Consider the rational inner function

\begin{equation} \phi(z_1,z_2)=-\frac{\tilde{p}(z)}{p(z)}=-\frac{4z_1z_2^2 - z_2^2 - 3z_1z_2 - z_2 + z_1}{4 - z_1 - 3z_2 - z_1z_2 + z_2^2},\label{AMYRIF}\tag{1.1} \end{equation}

which features as an example in the paper [3.3], and has been further studied in [3.9][Section 1.3] and [3.6][Example 5.2] (without the minus sign in front).

The corresponding RISP \(\Phi(z,w) = (\phi, w)\) has a SF point corresponding to a normal crossing of invariant components and a rotation band.

(a) \(\Phi^n\) for \(n=1\)
(b) \(\Phi^n\) for \(n=2\)
(c) \(\Phi^n\) for \(n=5\)
(d) \(\Gamma(\Phi)\cap\mathbb{T}^2\text{.}\) Parabolic fibers in pink.
Figure 1.4. Iteration of \(\Phi\) given by (1.1) on \(\mathbb{T}^2\text{.}\) (Successive images of the vertical axis marked red)

Subsection 1.2 \((1, n)\) RISPs with more complex behavior

Consider the function

\begin{equation*} \Phi(z,w)=\left(\frac{\left(13 w^4+8 w^3+10 w^2+1\right) z-(w-1)^2 \left(3 w^2+2 w+3\right)}{w^4-(w-1)^2 \left(3 w^2+2 w+3\right) z+10 w^2+8 w+13}, w\right). \end{equation*}

The invariant curves and level curves are

Figure 1.6. Invariant curves for \(\phi\)
Figure 1.7. Level curves for \(\phi\)
Notice that the level curves indicate that \(\Phi\) has a singular fixed point at \((z, w) = (1, -1)\)

The dynamical behavior for several iterations of \(\Phi\) is

(a) Vertical lines before mapping
(b) \(\Phi^n\) for \(n=1\)
(c) \(\Phi^n\) for \(n=2\)
(d) \(\Phi^n\) for \(n=3\)
Figure 1.8. Iteration of \(\Phi\)

Because there are three horizontal invariant curves, we should observe four “stacked” rotation belts bounded by identity fiber maps. A computation of the multipliers as a function of \(w\) gives the following visualization.

Figure 1.9. Rotation number as a function of fiber height

Consider the function

\begin{equation*} \Phi(z, w) = -\left(\frac{(3 - 4 i)- 10 w - (14 - 24 i) w^2 - (2 - 32 i) w^3 + (7 +12 i) w^4 + ((1 - 4 i) - 8 w - (4 - 24 i) w^2 + (12 + 32 i) w^3 + (15 + 12 i) w^4) z}{(15 - 12 i) + (12 - 32 i) w - (4 + 24 i) w^2 - 8 w^3 + (1 + 4 i) w^4 + ((7 - 12 i) - (2 + 32 i) w - (14 + 24 i) w^2 - 10 w^3 + (3 + 4 i) w^4) z},w\right). \end{equation*}

The invariant curves and level curves are

Figure 1.11. Invariant curves for \(\phi\)
Figure 1.12. Level curves for \(\phi\)
Notice that the level curves indicate that \(\Phi\) has singular fixed point at \((z, w) = (-1, -1)\) and \((z, w) = (1, -3/5 - 4/5i)\) and that each of these is a crossing of invariant curves.

The dynamical behavior for several iterations of \(\Phi\) is

(a) Vertical lines before mapping
(b) \(\Phi^n\) for \(n=1\)
(c) \(\Phi^n\) for \(n=2\)
(d) \(\Phi^n\) for \(n=3\)
Figure 1.13. Iteration of \(\Phi\)

With a choice of \(a = Pi\text{,}\) the polynomial \(Q_a\) factors as

\begin{equation*} 4 (w+1)^2 ((1+2 i) w+(-1+2 i))^2 \left((7-12 i) w^4+(12-8 i) w^3+26 w^2+(12+8 i) w+(7+12 i)\right)\text{,} \end{equation*}

exhibiting the normal crossings visible in the images.

Subsection 1.3 Higher order RISPs

Consider the function given by

\begin{equation*} \Phi(z, w) = \left(-\frac{-32 w^4 z^2+38 w^4 z-10 w^4+34 w^3 z^2-32 w^3 z+2 w^3-30 w^2 z^2+36 w^2 z-2 w^2+10 w z^2-8 w z-6 w-6 z^2+14 z-8}{8 w^4 z^2-14 w^4 z+6 w^4+6 w^3 z^2+8 w^3 z-10 w^3+2 w^2 z^2-36 w^2 z+30 w^2-2 w z^2+32 w z-34 w+10 z^2-38 z+32}, w\right), \end{equation*}

which has bidegree \((2, 4)\) as a RISP.

Unlike the \((1, n)\) case, invariant curves can now sweep through rotating fibers.

Figure 1.15. Invariant curves for \(\phi\)
Figure 1.16. Level curves for \(\phi\)
Notice that the level curves indicate that \(\Phi\) has singular fixed point at \((z, w) = (1, 1)\) and \((z, w) = (1, -1)\text{.}\)

One detail to note in the iteration portraits below is that the rotations are constrained by the fixed point structure of the initial vertical lines.

(a) Vertical lines before mapping
(b) \(\Phi^n\) for \(n=1\)
(c) \(\Phi^n\) for \(n=2\)
(d) \(\Phi^n\) for \(n=5\)
Figure 1.17. Iteration of \(\Phi\)

Consider the function \(\Phi\) with

\begin{equation*} \Phi(z, w) = \left(-\frac{2 w z-w-z}{-w-z+2},w^2\right), \end{equation*}

where we have \(\phi_2(w) = w^2\) in the second slot.

Even in this simple case, the behavior of \(\Phi\) is quite different from the related Example 1.1.

(a) Vertical lines before mapping
(b) \(\Phi^n\) for \(n=1\)
(c) \(\Phi^n\) for \(n=2\)
(d) \(\Phi^n\) for \(n=3\)
Figure 1.19. Iteration of \(\Phi\)

Consider the function \(\Phi\) with

\begin{equation*} \Phi(z, w) = \left(-\frac{2 w z-w-z}{-w-z+2},i w\right), \end{equation*}

where we have \(\phi_2(w) = i w\) in the second slot.

Again, the behavior of \(\Phi\) is quite different from the related Example 1.1.

(a) Vertical lines before mapping
(b) \(\Phi^n\) for \(n=1\)
(c) \(\Phi^n\) for \(n=2\)
(d) \(\Phi^n\) for \(n=5\)
Figure 1.21. Iteration of \(\Phi\)

We can observe the periodic structure of the mapping via an animation, where each frame represents the image of that iteration.

Figure 1.22. 10 iterations of \(\Phi\)

Subsection 1.4 Other two-variable examples

Consider the function

\begin{equation*} \Phi(z, w) = \left(z^2w^3, z^5w^2 \right). \end{equation*}
(a) Vertical lines before mapping
(b) \(\Phi^n\) for \(n=1\)
(c) \(\Phi^n\) for \(n=2\)
(d) \(\Phi^n\) for \(n=3\)
Figure 1.24. Iteration of \(\Phi\)

Let

\begin{equation*} B_a(z) = \frac{z - a}{ 1 - \cc{a}z}. \end{equation*}

Let \(\Phi\) be the RIM given by

\begin{equation*} \Phi(z,w) = (B_{1/2}(z)B_{i/3 }(w), B_{i/3 }(z) B_{-i/4 }(w)) \end{equation*}

Here, unlike the \((1, n)\) RISP case, we see chaotic behavior. In particular, we do not get curves of fixed points. The dynamical behavior for several iterations of \(\Phi\) is

(a) Vertical lines before mapping
(b) \(\Phi^n\) for \(n=1\)
(c) \(\Phi^n\) for \(n=2\)
(d) \(\Phi^n\) for \(n=3\)
Figure 1.26. Iteration of \(\Phi\)

Consider the function

\begin{equation*} \Phi(z, w) = \left(\exp \left(-\frac{w z+1}{1-w z}\right), w\right), \end{equation*}

with a so-called singular inner function in the first slot.

(a) Vertical lines before mapping
(b) \(\Phi^n\) for \(n=1\)
(c) \(\Phi^n\) for \(n=2\)
(d) \(\Phi^n\) for \(n=5\)
Figure 1.28. Iteration of \(\Phi\)