#### By Kevin M. Pilgrim

A Thurston map is a continuous, orientation-preserving, branched covering map $$f: S^2 \to S^2$$ of the two-sphere to itself, of degree at least two, with the property that the forward orbit of each of the finitely many branch points is finite. This latter condition means that over the complement of some finite set $$P$$, the iterates of $$f$$ are all unramified. In the mid-1980’s, W. Thurston showed that Thurston maps, up to a kind of homotopy-equivalence, could be used as combinatorial objects in the classification and characterization of rational maps $$g: \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$$ on the Riemann sphere, regarded as holomorphic dynamical systems [DH]. A Thurston map induces a pullback map on the Teichmüller space of an associated orbifold, and the map is equivalent to a rational map exactly when the pullback map has a fixed point. Nearly Euclidean Thurston (NET) maps, introduced in [CFPP], form an especially simple class. They are defined by the conditions that $$\#P=4$$ and that each branch point is simple. The associated Teichmueller space is a 2-dimensional disk, which is the simplest nontrivial possibility.

It turns out that an NET map admits a description in terms of a particular normal form, or NET presentation; an example is the parallelogram shown here. This is a very simple picture–equivalently, information you can store in a short text file. This connection with Euclidean geometry turns out to have remarkable dynamical consequences. NET maps are both more interesting and more tractable than we had expected.

For example, one of the basic invariants turns out to be a modular correspondence: a pair of holomorphic maps $$X, Y: \mathcal{W} \to \mathbb{H}^2/P\Gamma(2)$$, where $$Y: \mathcal{W}\to \mathbb{H}^2/P\Gamma(2)$$ is a finite cover of a famous modular surface familiar to number theorists. Group-theoretic invariants turn out to be related to affine symmetry groups of so-called {\em half-translation surfaces}–objects of intense recent study. Also, there are algorithms for the computation of many basic invariants, and for answering the three fundamental questions posed above. Intriguingly, according to the data we have to date, these algorithms appear to decide–in the formal, rigorous sense–the key question of equivalence to a rational map. The data also provide strong evidence for the conjecture that for a rational map $$f$$, the induced dynamics on free homotopy classes of simple closed curves in $$\widehat{\mathbb{C}} – P$$ obtained by taking preimages under $$f$$ has a finite global attractor; cf. [KPS, § 7]. Finally, the explicit nature of NET presentations allows for the algorithmic computation of associated wreath recursions on the fundamental group $$\pi_1(S^2-P)$$ induced by an NET map $$f$$—and this can be used as input in a recently developed software package [Bar] to confirm findings and, in the case when $$f$$ is equivalent to a rational map $$g$$, to approximate explicit formulas for $$g$$ and to draw its Julia set.

The time spent at AIM in our three SQuaRE workshops allowed us to patiently invest time and energy learning new mathematics from each other, and to develop the perspective and connections leading to the organization of the data and results in the website.

#### References

[Bar] Laurent Bartholdi. IMG, 2016. software package for GAP. https://github.com/ laurentbartholdi/img

[CFPP] J. W. Cannon, W. J. Floyd, W. R. Parry, and K. M. Pilgrim. Nearly Euclidean Thurston maps. Conform. Geom. Dyn. 16(2012), 209–255.

[DH] A. Douady and John Hubbard. A Proof of Thurston’s Topological Characterization of Rational Functions. Acta. Math. 171(1993), 263–297.

[KPS] Sarah Koch, Kevin M. Pilgrim, and Nikita Selinger. Pullback invariants of Thurston maps. Trans. Amer. Math. Soc. 368 No. 7(2016), 4621-4655.