Home Follow the rabbit Papers Software NET map documentation Thanks Gallery of pullback maps NET maps of degree 2 Dynamic portraits

Notes Nontrivial cycles Rationality degree 2 (16 portraits) degree 3 (94 portraits) degree 4 (272 portraits) degree 5 (144 portraits) degree 6 (338 portraits) degree 7 (152 portraits) degree 8 (476 portraits) degree 9 (153 portraits) degree 10 (353 portraits) degree 11 (153 portraits) degree 12 (483 portraits) degree 13 (153 portraits) degree 14 (353 portraits) degree 15 (153 portraits) degree 16 (483 portraits) degree 17 (153 portraits) degree 18 (353 portraits) degree 19 (153 portraits) degree 20 (483 portraits) degree 21 (153 portraits) degree 22 (353 portraits) degree 23 (153 portraits) degree 24 (483 portraits) degree 25 (153 portraits) degree 26 (353 portraits) degree 27 (153 portraits) degree 28 (483 portraits) degree 29 (153 portraits) degree 30 (353 portraits) degree 31 (153 portraits) degree 32 (483 portraits) degree 33 (153 portraits) degree 34 (353 portraits) degree 35 (153 portraits) degree 36 (483 portraits) degree 37 (153 portraits) degree 38 (353 portraits) degree 39 (153 portraits) degree 40 (483 portraits)

Hurwitz classes

Notes m=2, n=1 (4 classes) m=3, n=1 (9 classes) m=4, n=1 (24 classes) m=2, n=2 (10 classes) m=5, n=1 (25 classes) m=6, n=1 (88 classes) m=7, n=1 (47 classes) m=8, n=1 (133 classes) m=4, n=2 (85 classes) m=9, n=1 (120 classes) m=3, n=3 (43 classes) m=10, n=1 (269 classes) m=11, n=1 (140 classes) m=12, n=1 (618 classes) m=6, n=2 (201 classes) m=13, n=1 (228 classes) m=14, n=1 (583 classes) m=15, n=1 (646 classes) m=16, n=1 (789 classes) m=8, n=2 (503 classes) m=4, n=4 (155 classes) m=17, n=1 (469 classes) m=18, n=1 (1,544) m=6, n=3 (457) m=19, n=1 (629) m=20, n=1 (1,935) m=10, n=2 (621) m=21, n=1 (1,505) m=22, n=1 (1,902) m=23, n=1 (1,079) m=24, n=1 (3,976) m=12, n=2 (2,284) m=25, n=1 (1,678) m=5, n=5 (332) m=26, n=1 (3,037) m=27, n=1 (2,562) m=9, n=3 (1,032) m=28, n=1 (4,472) m=14, n=2 (1,384) m=29, n=1 (2,116) m=30, n=1 (8,521)

The zoo

Notes airplane corabbit rabbit Lodge's cubic Newton maps f_1/4 z² + i basilica mate basilica

A gallery of pullback maps

This is a hand-picked collection of pullback map drawings taken from the Hurwitz classes menu. They were chosen so that

deg(f) ≤ 9,
they are essentially different and
the group of all symmetries of the pullback map acts on the upper half-plane as a reflection group.

Each of these files contains a guess at the form of Thurston’s pullback map σ_f. It contains two views of the upper half-plane. The top view corresponds to the domain of σ_f, and the bottom view corresponds to the range. In the top view, black hyperbolic geodesics are the reflection axes of reflections in the list of EMOD generators given in filenameMOD.output. The same is true for the bottom view with “generators” replaced by “generator images”. The top view is the upper half-plane analog of filenameEMODTree.ps. It is a fundamental domain for the action of the subgroup of liftables in the extended modular group. The bottom view shows a guess at the image of this fundamental domain under σ_f.

NETmap labels axes of liftable reflections, and σ_f respects these labels. For example, a geodesic in the domain labeled “A” maps to the geodesic in the image labeled “A”.

Brackets underneath the real axis indicate folds. For more on these, see Section 3.10 of the file NETmap.pdf. Reflection axes of extra symmetries (for which, see the end of Section 3.2 of NETmap.pdf) which are reflections are drawn in color, as are their images under σ_f, and σ_f respects colors.

Condition 3) above makes it very likely that the drawing is correct. It seems to always be possible to verify that a drawing is correct by (verifying and) using the information in the Hurwitz classes menu.