Straight-Line Flow#

Acting on surfaces by matrices.#

from flatsurf import translation_surfaces

s = translation_surfaces.veech_double_n_gon(5)

s.plot()
../_images/e18fcc9eea1e1e4457039ea9253c2bbdf7aad32247b44af0dc59ce6effe6719b.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/249894febae957713f28a4913b852c71546acea95846391017d99412fbaf83e5.png

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

traj.flow(100)

s.plot() + traj.plot()
../_images/00edd74908db8a03e5bed97202fe61ee7e2fcb6f6ce6feb7bf7afb1e43dd9d0a.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/47470aec7bf660a20a4d52ecb87424961808886282299c515b32a9db0718596c.png

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