A Path Through the Roots

Paths have a logic and a mystery all their own. Milwaukee Avenue, which diagonally skewers the otherwise largely Cartesian Chicago layout, closely follows a Native American trail, used for centuries by the indigenous inhabitants of Illinois, which may perhaps have been first created by animals seeking the waters of Lake Michigan.

Figure 1.

A path-connected planar set.

Topologists care about paths and path-connectedness. And it sometimes happens that a topological space of interest can be shown to be path-connected in a constructive way, by exhibiting an explicit path between two given points.

Iterated function systems (hereafter IFSs) are a well-studied class of dynamical systems. One fixes a complete metric space $X$ and a finite collection of uniform contracting maps $f_1,\cdots , f_n$ of $X$, and considers the semigroup $S$ generated by the $f_i$. Associated to an IFS is its attractor $\Lambda$, the unique closed nonempty subset of $X$ with the property that $\Lambda$ is equal to the union of its images $f_i\Lambda$. Because of the uniform contraction of the $f_i$, one may obtain a concrete description of $\Lambda$ as the closure of the set of points $w(p)$ where $p \in \Lambda$ is arbitrary (e.g., we could take $p$ to be the unique fixed point of $f_1$) and where $w$ ranges over the elements of $S$.

Perhaps the simplest nontrivial examples of IFSs are obtained by taking $X=\mathbb{C}$, and considering the IFS generated by a pair $f,g$ of linear contractions with distinct fixed points and common contraction factor $c$ (with $0<|c|<1$). Up to conjugacy, we may normalize the generators as $f:z \to cz$ and $g:z \to 1+cz$ and in this way obtain a natural family of attractors $\Lambda (c) \subset \mathbb{C}$ parameterized by the punctured open unit disk $\mathbb{D}^*$. Figure 2 shows some examples of $\Lambda (c)$ for various $c$. In each example, $f\Lambda (c)$ is in blue and $g\Lambda (c)$ is in orange, and one can see quite viscerally how $\Lambda$ is obtained from the union.

Figure 2.

Attractors $\Lambda (c)$ for various $c$.

For this family of IFSs there is a clean dichotomy for $\Lambda (c)$: either $\Lambda (c)$ is connected (in fact, path-connected) or it is a Cantor set. In 1981 Barnsley–Harrington proposed the study of the “Mandelbrot-like” set $\mathcal{M}\subset \mathbb{D}^*$ consisting of parameters $c$ for which $\Lambda (c)$ is connected. The set $\mathcal{M}$ is depicted in Figure 3.

In 1988 Thierry Bousch, in an unpublished manuscript, proved that $\mathcal{M}$ is connected path-connected. His proof is rather elegant, and gives rise to explicit paths with their own particular charm, and we reproduce it in what follows.

The first (rather elementary but useful) observation is that $\mathcal{M}$ contains the entire annulus $\lbrace 1>|z|>1/\sqrt {2} \rbrace$. Here is a sort of heuristic argument, using Lebesgue measure. Let’s suppose the (2-dimensional) Lebesgue measure $\mu$ of $\Lambda (c)$ is positive (it is evidently finite). A linear contraction by the factor $|c|$ multiplies Lebesgue measure by $|c|^2$, so if $f\Lambda (c)$ and $g\Lambda (c)$ are disjoint, then from $\Lambda = f\Lambda (c) \cup g\Lambda (c)$ it follows that $\mu = 2|c|^2\mu$, or $|c|^2 = 1/2$. This suggests (and it turns out to be true) that for $|c|<1/\sqrt {2}$ the Lebesgue measure of $\Lambda (c)$ is zero, and for $|c|>1/\sqrt {2}$ the images $f\Lambda (c)$ and $g\Lambda (c)$ intersect nontrivially (in a set of positive Lebesgue measure).

Figure 3.

The set $\mathcal{M}$.

To go beyond this we look at $\mathcal{M}$ in another way. One very pleasant reason to study $\mathcal{M}$ is that it has, perhaps unexpectedly, an entirely different characterization from the dynamical one given above in terms of roots. We explain how. First, notice that $0$, being the unique fixed point of $f$, is necessarily contained in $\Lambda (c)$, and in fact $\Lambda (c)$ is the closure of the images $w(0)$ as $w$ ranges over the set of finite words in the alphabet $\lbrace f,g\rbrace$. If $w$ is such a word, the image $w(0)$ is the value of a polynomial $p_w$ evaluated at $c$, where the degree of $p_w$ is (at most) one less than the length of $w$, and the coefficients of $p_w$ (from lowest to highest) are $0$ or $1$ according to whether the letters of $w$ are $f$ or $g$.

This is best illustrated by example:

$$\begin{align*} gffgfg(0) &= gffgf(1) \\ &= gffg(c) = gff(1+c^2) = gf(c+c^3)\\ & = g(c^2+c^4)=1+c^3+c^5 \end{align*}$$

Words beginning with $f$ correspond to polynomials with coefficients in $\lbrace 0,1\rbrace$ and with constant term $0$, and words beginning with $g$ correspond to polynomials with coefficients in $\lbrace 0,1\rbrace$ and with constant term $1$. Taking closures, one sees that $\Lambda (c)$ is the set of values of power series with all coefficients in $\lbrace 0,1\rbrace$ evaluated at $c$.

It is not hard to show that $\Lambda (c)$ is connected if and only if $f\Lambda \cap g\Lambda$ is nonempty; equivalently if there are two power series $\phi ,\theta$ with coefficients in $\lbrace 0,1\rbrace$, where $\phi$ has constant term $1$ and $\theta$ has constant term $0$, such that $\phi (c)=\theta (c)$. Equivalently, $c$ is a root of the power series $(\phi -\theta )/2$ which has coefficients in $\lbrace -1,0,1\rbrace$. We finally arrive at the characterization of $\mathcal{M}$ as the set of $0<|c|<1$ which are roots of power series in one variable $z$ and coefficients in $\lbrace -1,0,1\rbrace$ (starting with constant term $1$ for concreteness).

OK, now let’s show $\mathcal{M}$ is path-connected. Let $\mathfrak{P}$ denote the set of power series in $z$ (with constant term $1$) and coefficients in $\lbrace -1,0,1\rbrace$ and let $\mathfrak{H}$ denote the space of closed subsets of the open unit disk (in the Hausdorff topology). There is a map $\rho :\mathfrak{P}\to \mathfrak{H}$ which takes a power series to its set of roots. We may topologize $\mathfrak{P}$ in an obvious way as a Cantor set and then ask about the continuity of $\rho$.

How do the roots of a power series depend on its coefficients? Since the coefficients are all bounded, one may show (e.g., by Rouché’s theorem) that for every $\epsilon$ and every $\lambda < 1$ there is an $n$ so that if $\phi ,\psi \in \mathfrak{P}$ agree on the first $n$ coefficients, then $\rho (\phi )$ and $\rho (\psi )$ are $\epsilon$-close in the Hausdorff distance when restricted to the disk of radius $\lambda$ (this is even true in the sense of counting roots with multiplicity). Since $\mathcal{A}\subset \mathcal{M}$ we only need to consider roots with absolute value at most $1/\sqrt {2}$, and therefore (in this range) $\epsilon$ may be controlled uniformly by $n$, and $\rho$ is (uniformly) continuous.

Using this observation we now indicate why $\mathcal{M}$ is path-connected. Let $\phi \in \mathfrak{P}$. For every $n$ we will construct a finite sequence of power series

$$\phi = B_0, A_1, B_1, \cdots , A_m = 1$$

where each $A_j$ ($j>0$) is a polynomial of degree $\le n$ obtained by truncating $B_{j-1}$, throwing away all terms of degree $> n$; and where

$$B_j = \frac{A_j}{1+\alpha (j)z^{n(j)}} \quad (j>0)$$

where $n(j)$ is the degree of $A_j$, and $-\alpha (j)$ is the coefficient of $z^{n(j)}$ in $A_j$.

Figure 4.

These two figures show approximate paths in $\mathcal{M}$ in degrees $n=16$ and $n=32$ from $-0.108715+0.78504i$ (one of the roots of $1+z^2-z^4+z^5-z^7+z^8-z^9+z^{11}-z^{12}-z^{13}-z^{15}$ of smallest absolute value) to $i$. The limit as $n \to \infty$ is an honest path in $\mathcal{M}$.

Since $1+\alpha (j)z^{n(j)}$ is nonzero throughout the open unit disk, $B_j$ has the same roots as $A_j$. Furthermore, since $A_j$ and $B_{j-1}$ agree on the first $n$ coefficients, $\rho (A_j)$ is $\epsilon$-close to $\rho (B_{j-1})$ outside the annulus $\mathcal{A}$. Since $A_m$ has no roots outside this annulus (in fact no roots at all) this gives a sequence of successive points in $\mathcal{M}-\mathcal{A}$, each at most distance $\epsilon$ from the previous one, connecting any root of $\phi$ to $\mathcal{A}$.

Finally, observe that increasing $n$ gives a new sequence that refines the old, where intermediate terms in the new sequence between successive terms in the old sequence all agree in degree $<n$.

This raises an interesting question. Recall that $\mathfrak{P}$ is our notation for the (Cantor) set of power series, and let $\mathfrak{E}$ denote the total space of the bundle over $\mathfrak{P}$ whose fiber over each $p\in \mathfrak{P}$ is its set of roots in the open unit disk. We can define an equivalence relation on $\mathfrak{E}$ which identifies the fibers over different $p,p'$ with the same roots in the open unit disk. Let $X$ denote the quotient space. Projecting to the unit disk defines a surjection $X \to \mathcal{M}$; since $X$ is path-connected, so is $\mathcal{M}$. What is the topology of $X$?

Author’s Note

Figure 1 was generated by the program ai-draw.tokyo. Bousch’s unpublished manuscript “Paires de Similitudes Z->SZ+1, Z->SZ-1” may be found online at https://www.imo.universite-paris-saclay.fr/~thierry.bousch/preprints/paires_sim.pdf.

Danny Calegari is a professor of mathematics at the University of Chicago. His email address is dannyc@math.uchicago.edu.

Article DOI: 10.1090/noti2738

Credits

Figure 1 is courtesy of and generated by the program ai-draw.tokyo.

Figures 2–4 and photo of Danny Calegari are courtesy of Danny Calegari.