Straight-Line Flow

Acting on surfaces by matrices.

from flatsurf import translation_surfaces

s = translation_surfaces.veech_double_n_gon(5)

s.plot()
../_images/cbd1ccf86cceae119014cb5b385e37fea9d49663a746407f41f2582095904734.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/89a89b932b8b13d2b895095caba96f0c6dacf75958afaca86e2faee999a04d0d.png

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

traj.flow(100)

s.plot() + traj.plot()
../_images/d3eeba029d6ef40d05049a44cc41463bb56f1eaa818e1fd8ceaf2e3a6fed3708.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/5c41eaf437b1214fa2c3a64170a5b86b6d5534954dc1517a2e9872ab106d67bb.png

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