Numerical Results

Abstract

We develop a computational scheme for the parabolic renormalization operator which is based on the asymptotics of the Fatou coordinates at infinity, and apply it to numerical computations of the basin and the domain of the renormalization fixed point and of the spectrum of the parabolic renormalization operator at the fixed point.

4.1 A Computational Scheme for $\fancyscript{P}$

Fig. 4.1

The domain of analyticity of $\fancyscript{P}f_0(z)$ for $f_0(z)=z+z^2$, with the immediate parabolic basin indicated

Having mentioned the resurgent properties of the asymptotic expansion of the Fatou coordinate, we proceed to describe the computational scheme for $\fancyscript{P}$ (see Fig. 4.1). We begin with a germ of an analytic mapping

$$f(z)=z+z^2+O(z^3)$$

defined in a neighborhood of the origin. Applying the change of coordinates $w=\kappa (z)=-1/z$, we obtain

$$F(w)=w+1+\frac{A}{w}+O\left( \frac{1}{w^2}\right) $$

defined in a neighborhood of $\infty$. We again use the notation $\varPhi _A(w)$ for the function that conjugates $F$ with the unit translation

$$\varPhi _A(F(w))=\varPhi _A(w)+1$$

for $\mathrm{Re}\,w\gg 1$. We let $\varPhi _R(w)$ be the solution of the same functional equation for $\mathrm{Re}\,w\ll -1$. These changes of coordinate are well-defined up to an additive constant, and

$$\phi _A(z)=\kappa ^{-1}\circ \varPhi _A\circ \kappa (z),\quad \phi _R(z)=\kappa ^{-1}\circ \varPhi _R\circ \kappa (z).$$

As we have seen in Theorem 2.2, the function $\varPhi _A(w)$ has an asymptotic development

$$\varPhi _A(w)\sim w- A\log w + \text {const}_A +\sum _{k=1}^\infty b_k w^{-k}.$$

The coordinate $\varPhi _R(w)$ has an identical asymptotic development, differing only by the value of $\text {const}_R$. While this may seem surprising at first glance, recall that these functions are Laplace transforms of different analytic continuations of the Borel transform of the same divergent series (plus the $w-A\log w+ \text {const}$ term).

We select a large integer $M$ (in practice, $M\approx 100$). We will use the asymptotic expansion to estimate $\varPhi _A(w)$ for $w\ge M$ and $\varPhi _R(w)$ for $w\le -M$. Consider an iterate $N\approx 2M$ such that

$$\mathrm{Re}\,F^N(w)\ge M \quad \text { for }\mathrm{Re}\,w\in [-M-1,-M].$$

Let $v(z)$ be the function

$$\nu (z) = \mathrm{ixp}\circ \varPhi _A \circ F^N\circ (\varPhi _R)^{-1}\circ \mathrm{ixp}^{-1}(z).$$

It differs from the parabolic renormalization $\fancyscript{P}(f)$ only by rescaling the function and its argument:

$$\fancyscript{P}(f)(z)=a_1\nu (a_0 z).$$

Now consider a contour $\varGamma$ connecting $w=-M-1+\textit{iH}$ with $F(w)\approx -M+\textit{iH}$ which is mapped onto the circle $S_\rho =\{|z|=\rho \}$ for a small value of $\rho$ by $\mathrm{ixp}\circ \varPhi _R$. Select $n\in {\mathbb N}$ and consider the $n$ points in $S_\rho$ given by $x_k=\rho \exp (2\pi k/n),\; k=0,\ldots ,n-1$. We then evaluate the first $n$ coefficients in the Taylor expansion of $\eta$ at the origin

$$\eta (z)=\sum _{j=0}^\infty r_j z^j$$

using a discrete Fourier transform. Specifically, we calculate

$$s_k=\nu (x_k)\approx \sum _{j=0}^{n-1}r_j (x_k)^j=\sum _{j=0}^{n-1}r_j\rho ^j \exp (2\pi kj/n),$$

and apply the inverse discrete Fourier transform:

$$r_j\approx \frac{1}{n\rho ^j}\sum _{k=0}^{n-1} s_k \exp (- 2\pi kj/n).$$

Since

$$\fancyscript{P}(f)(z)=\sum _{j=1}^\infty s_ja_1a_0^jz^j,$$

we have

$$a_1a_0s_1=1,\text { and further }a_0=\frac{s_1}{s_2}.$$

This step completes the computation of the Taylor expansion of $\fancyscript{P}(f)$.

4.1.1 Computing $f_*$

In computing the fixed point $f_*(z)$ we find it more convenient to work with the representation of a germ $f(z)=z+z^2+\cdots$ in the form

$$f(z)=z\exp (f_{\log }(z)),$$

where $f_{\log }$ is a germ of an analytic function at the origin with $f_{\log }(z)=z+\cdots$. We then rewrite the parabolic renormalization operator in terms of its action on $f_{\log }$:

$$\fancyscript{P}_{\log } (f_{\log })(z)=(2\pi i)^{-1}\varPhi _A\circ F^{N}\circ (\varPhi _R)^{-1}\circ \mathrm{ixp}^{-1}(z)-\mathrm{ixp}^{-1}(z).$$

This helps to avoid the round-off error which arises from the growth of $f_*$ near the boundary $\partial \mathrm{Dom }(f_*).$

Modifying the scheme described above for the operator $\fancyscript{P}_{\log }$, we calculate the fixed point by iterating $\fancyscript{P}$ starting at $f_0(z)=z+z^2$:

Empirical Observation 4.1

$$f_*(z)\approx z+z^2+0.(514 -0.0346i)z^3+\cdots .$$

Our calculations appear reliable up to the size of the round-off error in double-precision arithmetic ($\sim$ $10^{-14}$) in the disk of radius $r=5$ around the origin. As we will see below, the true radius of convergence for the series for $f_*$ is approximately $41$ (see the Empirical Observation 4.4).

We also estimated the leading eigenvalue of $D\fancyscript{P}|_{f_*}$:

Empirical Observation 4.2

The eigenvalue of $D\fancyscript{P}|_{f_*}$ with the largest modulus is

$$\lambda \approx -0.017+ 0.040i,\quad |\lambda |\approx 0.044.$$

The small size of $\lambda$ explains the rapid convergence of the iterates of $\fancyscript{P}$ to the fixed point. To obtain this estimate, we write

$$f(z)=z+z^2+\sum _{k=3}^\infty \text {coeff}_k(f) z^k,$$

and consider the spectrum of the $N\times N$ matrix $A=(a_{ij})_{i,j=3 \ldots N+3}$, with

$$a_{ij}=\frac{\text {coeff}_j(\fancyscript{P}(f_*+\varepsilon z^i))-\text {coeff}_j(f_*)}{\varepsilon },$$

which serves as a finite-dimensional approximation to $D\fancyscript{P}|_{f_*}$.

4.2 Computing the Domain of Analyticity of $f_*$

4.2.1 Computing the Tail of the Domain $\text {Dom}(f_*)$

Computing the tail using an approximate self-similarity near the tip Let us denote

$$\begin{aligned} {t_{*}}\equiv {t^{f_*}}=\partial \text {Dom}(f_*)\cap \overline{B_0^{f_*}} \end{aligned}$$

the endpoint of the tail of the immediate basin of $f_{*}$. Let $C_R$ be a repelling fundamental crescent of $f_{*}$, and let $w\in C_{R}$ have the property

$$t_*=\mathrm{ixp}\circ \phi _R(w).$$

Let $k\ge 2$ be such that

$$\begin{aligned} f^k_{*}(w)=0,\text { so that }f^{k-1}_{*}(w)=t_{*}. \end{aligned}$$

Denote $\chi$ the local branch of $f_{*}^{-(k-1)}$ which sends $t_*$ to $w$. Then the composition

$$\begin{aligned} \nu \equiv \mathrm{ixp}\circ \phi _R\circ \chi \end{aligned}$$

is an analytic map defined in a neighborhood of the endpoint $t_{*}$, which fixes it:

$$\begin{aligned} \nu (t_{*})=t_{*}. \end{aligned}$$

This point can be found numerically:

Empirical Observation 4.3

$$t_*\approx -779.306-643.282i, \,\mathrm {and}\, \nu '(t_*)\approx 0.232+0.264i.$$

Thus, we have identified the endpoint of the largest tail of $\text {Dom}(f_*)$. This construction also gives us the means to compute the tail itself. This can be done by successively applying $\nu$ to the immediate basin $B_0^{f_*}$, thus pulling it in towards $t_*$.

Now let $q\in C_R$ be any other preimage of $0$:

$$f_*^l(q)=t_*\quad \text { for some } q\in {\mathbb N}.$$

Then $v=\mathrm{ixp}\circ \phi _R(q)$ is the endpoint of a different tail in $\partial \text {Dom}(f_*)$. It can be computed by first pulling back the tail of $B_0^{f_*}$ using the inverse branch

$$f_*^{-l}:t_*\mapsto q,$$

and then applying $\mathrm{ixp}\circ \phi _R.$

Computing the tail using the functional equation for an inverse branch A more careful analysis of the tail can be done as follows (Fig. 4.2). Denote by $\xi$ the local branch of $f_*^{-1}$ defined in a slit neighborhood $D_r(0)\setminus [0,r)$ for some small value of $r,$ that sends $0\mapsto t_*$. We can write the renormalization fixed point equation for this particular branch:

$$\begin{aligned} \xi =\psi _R\circ \xi \circ \psi _A^{-1}, \end{aligned}$$

(4.1)

where $\psi _R=\chi \circ \mathrm{ixp}\circ \phi _R$, and $\psi _A^{-1}$ is the appropriately chosen branch of $(\mathrm{ixp}\circ \phi _A)^{-1}$ (thus the “self-similarity” of the tail is exponential, rather than linear). We are going to use the renormalization equation (4.1) inductively to compute $\xi (z)$ for sufficiently small values of $z$, and thus plot the tail.

Fig. 4.2

The inverse branches used in computing the tail of $\text {Dom}(f_*)$

Representing the numbers in the image of the tail Numerical computations indicate that the value of $r=0.0002$ is sufficiently small for our needs, and for $|z|<r$ the difference between the left and the right sides of (4.1) is of the order of $10^{-11}$. The values of $z$ for which we would like to evaluate $\xi (z)$ become too small to be represented by the standard double precision numbers (and even too small for their logarithms to be so represented). We write

$$s(t)=\exp (2\pi t),$$

and choose $\hat{t}$ so that

$$\exp (-2\pi \hat{t})=0.0002,\quad \text { that is }\hat{t}=1.3555\ldots .$$

We then represent a small positive number $x$ as

$$x=\frac{1}{s^k(t)},$$

for the unique choices of $t\in [\hat{t},s(\hat{t})),$ and an iterate $k\in {\mathbb N}$.

Fig. 4.3

The domain of analyticity of $f_*$ and the boundary of the immediate parabolic basin $B_0^{f_*}$. In the second figure, a part of the critical level curve of $f_*$ is also indicated

Fig. 4.4

A blow-up of the boundary of the immediate basin of $f_*$ in the vicinity of the parabolic point

We can write any complex number $z$ with $|z|<r$ uniquely as

$$z=(k,t,\theta )\equiv \frac{\exp (2\pi i\theta )}{s^k(t)},\quad 0\le \theta <1.$$

Note that this representation of small numbers makes it very easy to compute logarithms. In particular,

$$\mathrm{ixp}^{-1}((k,t,\theta ))=\theta + \textit{is}^{k-1}(t).$$

The next step in applying (4.1) is to apply $\phi _A^{-1}$ to the right-hand side of the equation. From the first two terms in the asymptotics of

$$\phi _A(z)=-\frac{1}{z}+O(\log |z|)\quad \text { for small }z,$$

it follows that

$$\phi _A^{-1}(y)=-\frac{1}{y+O(\log |y|)}\quad \text { for large }|y|.$$

A numerical estimate shows that for $|y|\ge 10^{18}$, the $O(\log |y|)$ term dissapears into the round-off error when added to $y$. Thus

$$\psi _A^{-1}((k,t,\theta ))\approx -\frac{1}{\theta +i s^{k-1}(t)}\approx \textit{is}^{k-1}(t)=(k-1,t,1/4),$$

provided $s^{k-1}(t)\ge 10^{18}.$ A direct estimate shows that for either $k\ge 3$, or $k=2$ and $t>18\log 10/2\pi \approx 6.596,$ the last inequality will hold.

The size of the domain of analyticity To draw the pictures of the domain of analyticity of the fixed point of $f_*$ (Figs. 4.3 and 4.4) we employed the following strategy. First, a periodic orbit of period $2$ in $\partial B_0$ was identified. Its preimages give a rough outline of $\partial B_0$, but become sparse near the “tails”, which are not visible in this initial outline. At the next step, the large “tail” of $B_0$ is computed as described above. Finally, its preimages are used to fill in the remaining gaps in $\partial B_0$.

As the final step, we calculate the boundary of $\mathrm{Dom }(f_*)$ as

$$\partial \mathrm{Dom }(f_*)=\mathrm{ixp}\circ \phi _R(\partial B_0\cap P_R).$$

An empirical estimate of the inner radius of $\text {Dom}(f_*)$ around the origin allows us to formulate the following observation (see Fig. 4.3):

Empirical Observation 4.4

The radius of convergence of the Taylor expansion of $f_*$ at the origin is $R\approx 41.$