Postcritically Finite Maps
Dimension 1, degree 4 polynomials over ℚ
A map is postcritically finite (PCF) if every critical point has a finite forward orbit. Its critical portrait is the graph of the critical points and their forward orbits. Critical points are counted over the algebraic closure, including ∞ for polynomials.
Functions over ℚ: 16 conjugacy classes over Export to Sage
| Label | Function | Field of Definition | Critical Points | Critical Point Field | Portrait Size | Portrait Cycles | Portrait Components | Portrait | Citations |
|---|---|---|---|---|---|---|---|---|---|
| 1.880302df.903704ba.1 | (x4 − 4x2y2 + 2y4 : y4) | 1.1.1.1 | 4 | 2.2.8.1 | 6 | 1, 1 | 5, 1 | view | Fraser2024 |
| 1.e029f05a.0d52105e.2 | (x4 − 4x2y2 : 2y4) | 1.1.1.1 | 4 | 2.2.8.1 | 5 | 1, 1 | 4, 1 | view | Fraser2024 |
| 1.5457b5e3.e9909514.2 | (x4 − 4x2y2 + 4y4 : 2y4) | 1.1.1.1 | 4 | 2.2.8.1 | 5 | 1, 1 | 4, 1 | view | Fraser2024 |
| 1.5457b5e3.e9909514.1 | (2x4 − y4 : -y4) | 1.1.1.1 | 2 | 1.1.1.1 | 4 | 1, 1 | 3, 1 | view | Fraser2024 |
| 1.8af865e4.2d8ff491.1 | (x4 − 2/3·x2y2 − 8/27·xy3 + 26/27·y4 : y4) | 1.1.1.1 | 3 | 1.1.1.1 | 4 | 1, 1, 1 | 2, 1, 1 | view | Fraser2024 |
| 1.9da62374.0f726d75.1 | (x4 − 2/3·x2y2 + 8/27·xy3 − 19/27·y4 : y4) | 1.1.1.1 | 3 | 1.1.1.1 | 4 | 2, 1 | 3, 1 | view | Fraser2024 |
| 1.a70750b0.c07be85c.1 | (3x4 + 4x3y − 6x2y2 − 12xy3 + 5y4 : 6y4) | 1.1.1.1 | 3 | 1.1.1.1 | 4 | 1, 1 | 3, 1 | view | Fraser2024 |
| 1.da89c05a.4e01541d.1 | (3x4 − 8x3y : -6y4) | 1.1.1.1 | 3 | 1.1.1.1 | 4 | 1, 1 | 3, 1 | view | Fraser2024 |
| 1.770c2622.7cef54b3.2 | (x4 − 2x2y2 + y4 : y4) | 1.1.1.1 | 4 | 1.1.1.1 | 4 | 2, 1 | 3, 1 | view | Fraser2024 |
| 1.e029f05a.0d52105e.1 | (2x4 − 4x2y2 + y4 : y4) | 1.1.1.1 | 4 | 1.1.1.1 | 4 | 1, 1 | 3, 1 | view | Fraser2024 |
| 1.6951d87f.f9bbe0ed.1 | (x4 − 2x2y2 : y4) | 1.1.1.1 | 4 | 1.1.1.1 | 4 | 1, 1, 1 | 2, 1, 1 | view | Fraser2024 |
| 1.8e57877c.24b4743f.1 | (x4 + 4xy3 : 3y4) | 1.1.1.1 | 4 | 2.0.3.1 | 4 | 1, 1, 1, 1 | 1, 1, 1, 1 | view | Fraser2024 |
| 1.770c2622.7cef54b3.1 | (x4 − y4 : y4) | 1.1.1.1 | 2 | 1.1.1.1 | 3 | 2, 1 | 2, 1 | view | Fraser2024 |
| 1.d445c2f7.72c00a36.1 | (x4 : 4xy3 + 3y4) | 1.1.1.1 | 3 | 1.1.1.1 | 3 | 1, 1, 1 | 1, 1, 1 | view | Fraser2024 |
| 1.a48ef78e.703b5deb.1 | (3x4 − 4x3y + y4 : y4) | 1.1.1.1 | 3 | 1.1.1.1 | 3 | 2, 1 | 2, 1 | view | Fraser2024 |
| 1.024ac6d7.76c4de71.1 | (x4 : y4) | 1.1.1.1 | 2 | 1.1.1.1 | 2 | 1, 1 | 1, 1 | view | FN1997, GHJSX2021, Fraser2024 |