Unimodal map.
A unimodal map is a map $f$ from a compact interval $I$ to itself,
$$
f: I \to I,
$$
which is continuous, increasing on the first half of $I$ and decreasing on the second half, with
$$
f(\partial I) \subset \partial I.
$$
At the turning point $c$, the map may or may not be differentiable: in the differentiable case we have $f'(c)=0$ (as in the logistic family), while in the non-differentiable case (as in the tent family) the derivative does not exist at $c$.
Tent family
The tent map family $\{T_\lambda\}_{0\leq \lambda \leq 2}$ is defined on $[0,1]$ by
$$
T_\lambda(x) =
\begin{cases}
\lambda x, & 0 \leq x \leq \frac{1}{2}, \\[6pt]
\lambda (1-x), & \frac{1}{2} < x \leq 1,
\end{cases}
$$
where $\lambda$ is the slope parameter.
Logistic map family
The logistic map family $\{f_\mu\}_{0\leq \mu \leq 4}$ is defined on $[0,1]$ by
$$
f_\mu(x) = \mu\, x(1-x),
$$
where $\mu$ is the parameter controlling the slope at $x=0$ and the height of the maximum at $x=\frac12$.
Unimodal permutation
The turning point $c$ is periodic if there is $n\geq 1$ such that $f^n(c)=c$. The order in $\mathbb{R}$ induces a total order of the orbit $\mathcal{O}=\{x_0=x, x_1=f(c), \dots, ...x_{n-1}=f^{n-1}(c)\}$. Note that $x_i < x_1$ for every $i\neq 0$.
A unimodal permutation of period $n$ is a total order on $\{0,1,\dots, n-1\}$ such that the cyclic permutation $\pi(i) = i+1 \pmod n$ can be realized with this order in a real tent map. In other words, there exists a tent unimodal map $T_\lambda$ on $[0,1]$ and an injective map
$$
h\colon \{0,1,\dots, n-1\} \to [0,1]
$$
that preserves the unimodal order, with $h(0)=c$ being the turning point, and $$h(\pi(j))=T_\lambda(h(j))$$ for every $j$.
The question
Given a unimodal permutation, is there a logistic map whose critical point is periodic and realizes this permutation?
How many parameters are there, and how can we find them?
The answer is:
- There is always one such parameter.
- This parameter is unique.
- There exist algorithms that allow us to find this parameter very efficiently.
After decades of research in complex dynamics, there are now many ways to prove these results.
I will present an idea of the original proof, due to W. Thurston (a Fields Medalist). It uses Teichmüler theory.
Spider map (concrete version of Thurston proof by Hubbard and Schleicher)
The algorithm to find the parameter is as follows. Consider a unimodal order $<$ on $\{0,1,\dots,n-1\}$.
Choose an arbitrary vector $(x_0,x_1,\dots,x_{n-1})\in [0,1]^n$ such that $x_0 = 1/2$ and
$$
x_i < x_j \quad \text{if and only if} \quad i < j
$$
in the given unimodal order.
Define the spider map $S$ on these $n$-tuples by
$$
(y_0,y_1,\dots,y_{n-1}) = S(x_0,x_1,\dots,x_{n-1})
$$
as follows.
Choose the parameter $\mu \in [0,2]$ in the logistic family such that
$$
f_\mu(1/2) = x_1.
$$
There is exactly one such parameter.
For each $i \neq 0$, define $y_i \in [0,1]$ as the unique point satisfying
$$
f_\mu(y_i) = x_{\,i+1 \pmod n},
$$
together with the condition
$$
y_i < 1/2 \ \text{if } x_i < 1/2, \quad
y_i > 1/2 \ \text{if } x_i > 1/2.
$$
Set $y_0 = 1/2$.
One can check that this new vector $(y_0,y_1,\dots,y_{n-1})$ also respects the given unimodal permutation.
It turn out we can observe that, no matter which initial vector $x = (x_0,x_1,\dots,x_{n-1})$ you choose,
$$
\lim_{k \to \infty} S^k(x) = z
$$
exists (the convergence is exponential) and does not depend on $x$. Indeed $z$ is the unique fixed point of $S$.
The logistic map whose critical value is $z_1$ is precisely the map whose critical point realizes the given unimodal permutation.
Why this happens?
Teichmüller Theory
Thurston’s argument is quite surprising: he complexifies the algorithm,
considering $n$ distinct complex points $z_0, z_1, \dots, z_{n-1}$
on the Riemann sphere $\overline{\mathbb{C}}$.
I am simplifying a bit, since one also needs to attach additional topological
information to describe the relative “position” of the points, in order to
replace the real-line order that is lost in the complex setting
(some “legs”, curves going from the points $x_k$ to $\infty$, so the collection of curves looks as a spider with very long legs as discussed in the article by Hubbard and Schleicher).
The key idea in Thurston’s approach is that, instead of interpreting
$(x_0,x_1,\dots,x_{n-1})$ as a vector, he views it as the hyperbolic Riemann surface
$$
\overline{\mathbb{C}}\setminus \{x_0,x_1,\dots,x_{n-1}\}
$$
with marked ends.
In this way, the complexified spider map is seen as a map between Riemann surfaces.
All these Riemann surfaces share the same topological type: a sphere $\mathbb{S}^2$ minus $n$ points.
Thurston then considers the Teichmüller space $\mathcal{T}$ of Riemann surfaces of the type
$$
\overline{\mathbb{C}} \setminus \{x_1, \dots, x_{n-1}\}.
$$
In this Teichmüller space, each point represents a Riemann surface of that type (more precisely, a conformal equivalence class of Riemann surfaces that are isotopic).
The Teichmüller space $\mathcal{T}$ is itself a complex-analytic manifold (in fact, a topological disk whose dimension depends on the number of removed points).
The spider map induces a complex-analytic map acting on the Teichmüller space
$$
\sigma: \mathcal{T} \to \mathcal{T}.
$$
Thurston proved that this map has a unique fixed point.
The argument uses the Teichmüller metric on $\mathcal{T}$, which makes $\mathcal{T}$ a complete metric space. One shows that $\sigma^2 = \sigma \circ \sigma$ is a weak contraction in this metric.
This is not enough by itself to guarantee a fixed point.
One also needs to prove that, for any starting point $s$ (represented by a Riemann surface), the sequence of iterates $\sigma^n(s)$ does not escape to the boundary of Teichmüller space. Equivalently, the hyperbolic geometry of the corresponding Riemann surfaces does not degenerate: their injectivity radii remain uniformly bounded away from zero, or, in other words, the lengths of simple closed geodesics on the surfaces representing $\sigma^n(s)$ remain controlled and do not blow up.
To be fair, this is highly sophisticated material, and I do not know all the details. Nevertheless, I have always found this idea truly astonishing.
Final comments
Teichmüller theory has been a major source of inspiration in complex dynamics. D. Sullivan, A. Douady & J. H. Hubbard, and R. Mañé, P. Sad & D. Sullivan were pioneers in applying quasiconformal methods to complex dynamics,
a development that profoundly transformed the field.
In the specific case of renormalization theory, numerous deep results by Sullivan, McMullen, Yoccoz, Lyubich, Wellington de Melo, and many others rely fundamentally on these methods.
References
Thurston never wrote down his proof.
A complete exposition can be found in the paper by Douady and Hubbard below. It deals with a far more general situation: rational maps with complex coefficients.
A. Douady and J. H. Hubbard, A proof of thurston's topological characterization of rational functions, Acta Mathematica, 171 (1993), 263-297.
This paper is less abstract, introducing the spider map and focuses on polynomials rather than rational functions:
J. H. Hubbard and D. Schleicher, The spider algorithm, Complex Dynamical Systems, RL Devaney ed., Proc. Symp. Appl. Math, 49 (1994), 155-180.
