Straight-Line Flow#

Acting on surfaces by matrices.#

from flatsurf import translation_surfaces

s = translation_surfaces.veech_double_n_gon(5)

s.plot()
../_images/b30c3e3e6a50b59bc3137a7fa61e333d908d0cdb57b03b119767c66ca53ee1eb.png

Defines the tangent_bundle on the surface defined over the base_ring of s.

TB = s.tangent_bundle()

baricenter = sum(s.polygon(0).vertices()) / 5

Define the tangent vector based at the baricenter of polygon 0 aimed downward.

v = TB(0, baricenter, (0, -1))

Convert to a straight-line trajectory. Trajectories are unions of segments in polygons.

traj = v.straight_line_trajectory()

s.plot() + traj.plot()
../_images/a2b7d325aa2130554fe032516c74b9556c2ce3bf90d85a14ef60eac6e0ab2382.png

Flow into the next \(100\) polygons or until the trajectory hits a vertex.

traj.flow(100)

s.plot() + traj.plot()
../_images/96478316e580ed860d43e3705c52efac4a48388c0528726afe815022c5a670d0.png

We can tell its type.

traj.is_saddle_connection()

True

You can also test if a straight-line trajectory is closed or a forward/backward separatrix.

Lets do it again but in the slope one direction.

v = TB(0, baricenter, (1, 1))

traj = v.straight_line_trajectory()

traj.flow(100)

s.plot() + traj.plot()
../_images/54e47cab6f26181dd95ddf26d557d9c3fc5c697c67edebfe1e4e8dc490337907.png

We remark that it follows from work of Veech that the slope one direction is ergodic for the straight-line flow.