Quadratic Thurston maps are postcritically finite branched coverings, which may be equivalent to a rational map or not. The talk focuses on maps with four postcritical points, that are covered by a real affine map on a lattice. So, many questions from complex dynamics are solved explicitly by linear algebra, including:
Is the map equivalent to a rational map?
How about essential matings from non-conjugate limbs?
When are two Thurston maps equivalent?
What is the effect of a Dehn twist?
Are preimages of an essential curve eventually periodic?
Is the virtual endomorphism of the pure mapping class group contracting?
