Straight-line flow

Initial version by Pat Hooper whooper@ccny.cuny.edu, Dec 16, 2017.



In [1]:

    
from flatsurf import *

Acting on surfaces by matrices.



In [2]:

    
s = translation_surfaces.veech_double_n_gon(5)



In [3]:

    
s.plot()









    Out[3]:

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



In [4]:

    
TB = s.tangent_bundle()



In [5]:

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

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



In [6]:

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

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



In [7]:

    
traj = v.straight_line_trajectory()



In [8]:

    
s.plot()+traj.plot()









    Out[8]:

Flow into the next $100$ polygons or until the trajectory hits a vertex.



In [9]:

    
traj.flow(100)



In [10]:

    
s.plot()+traj.plot()









    Out[10]:

We can tell its type.



In [11]:

    
traj.is_saddle_connection()









    Out[11]:





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.



In [12]:

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



In [13]:

    
traj=v.straight_line_trajectory()



In [14]:

    
traj.flow(100)



In [15]:

    
s.plot()+traj.plot()









    Out[15]:

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