Start with a simple dynamical system

Iteration means feeding the output of a function back into the same function.

For $f(x)=x^2-1$:

$$0\longmapsto -1\longmapsto 0\longmapsto -1\longmapsto\cdots\text{ and }\infty\to\infty$$

The critical points eventually repeat. Such a map is called post-critically finite, or PCF.

What can the infinitely many backward paths of this very simple system tell us?

Now run the process backward

Choose a point $t$. First solve $f(x)=t$, then solve $f^2(x)=t$, and continue.

$t \leftarrow f^{-1}(t) \leftarrow f^{-2}(t) \leftarrow \cdots$

The answers branch into a tree. The preimage of a loop from $t$ around a critical point is a path connect branches.

The iterated monodromy group records all of these shuffles. $$ \rho_f \colon \pi_1\!\left(\mathbb{P}^1(\mathbb{C})\setminus P_f,\,t\right) \longrightarrow \operatorname{Aut}(T_f), $$

Why introduce a spider?

We need a consistent way to draw one loop around each special point.

Draw non-crossing paths from one common base point. Travel out along one leg, circle its endpoint, and return along the same leg.

This collection of legs is a spider.

What is hard for a computer?

A person can often see when one path can be smoothly deformed into another.

Can a computer recognize the same path without continuously testing every possible deformation?

paths may bend, stretch, and look completely different;
they may not cross a special point or one another;
the method must also work near the point at infinity.

Key idea: replace the picture by a graph

Pull the spider backward through the function. The result is an embedded graph, which I call the web.

The web records only finite information:

which important points are connected;
which paths share a vertex;
the clockwise order of edges around each vertex.

The continuous deformation problem is now a finite graph-search problem.

The algorithm

1
Approximate each spider leg by finitely many points

2
Pull back the legs and build the web

3
Search for paths that do not conflict

4
Check every local neighborhood and its clockwise order

Output: a finite certificate that the selected paths form a good spider.

When does the computer say “good”?

Sufficient checks

The selected paths pass if:

they connect one chosen head to all required endpoints;
they do not cross or compete for an interior vertex;
the unused edge at each endpoint points to the correct special point;
the clockwise order agrees with the original spider.

Passing these checks guarantees that the lifted loops can be recognized automatically.

Example: $f(x)=2x^3-3x^2+1$

Pulled-back web

Compatible paths found by the search

Why this matters

Continuous deformation problem
$\Downarrow$
finite graph + path search + clockwise order
$\Downarrow$
a compact automaton describing infinitely many tree symmetries

Thank you!

Examples database