## Nearly Euclidean Thurston maps

###
J. W. Cannon, W. J. Floyd, W. R. Parry and K. M. Pilgrim

April 16, 2012

### Abstract

We take an in-depth look at Thurston's combinatorial characterization of
rational functions for a particular class of maps we call nearly
Euclidean Thurston maps. These are orientation-preserving branched maps
from the 2-sphere into itself whose local degree at every critical point
is 2 and which have exactly four postcritical points. These maps are
simple enough to be tractable, but are complicated enough to have
interesting dynamics.
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