Relative Period Deformations

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



In [1]:

    
from flatsurf import *

The Arnoux-Yoccoz surface



In [2]:

    
s = translation_surfaces.arnoux_yoccoz(3).canonicalize()



In [3]:

    
s.plot()









    Out[3]:



In [4]:

    
field=s.base_ring()
field









    Out[4]:





Number Field in alpha with defining polynomial x^3 + x^2 + x - 1



In [5]:

    
alpha = field.gen()
AA(alpha)









    Out[5]:





0.5436890126920763?



In [6]:

    
m=matrix(field,[[alpha,0],[0,1/alpha]])
show(m)









    




 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \alpha & 0 \\ 0 & \alpha^{2} + \alpha + 1 \end{array}\right)$

Check that $m$ is the derivative of a pseudo-Anosov of $s$.



In [7]:

    
(m*s).canonicalize()==s









    Out[7]:





True

Rel deformation

A singularity of the surface is an equivalence class of vertices of the polygons making up the surface.



In [8]:

    
s.singularity(0,0)









    Out[8]:





singularity with vertex equivalence class frozenset([(0, 0), (9, 1), (3, 0), (11, 2), (8, 0), (6, 0), (2, 1), (7, 2), (1, 2), (7, 0), (5, 2), (6, 1), (3, 1), (2, 2), (10, 0), (4, 1), (0, 2), (10, 2)])

We'll move this singularity to the right by two different amounts:



In [9]:

    
s1=s.rel_deformation({s.singularity(0,0):vector(field,(alpha/(1-alpha),0))}).canonicalize()



In [10]:

    
s2=s.rel_deformation({s.singularity(0,0):vector(field,(1/(1-alpha),0))}).canonicalize()



In [11]:

    
# Note that by the action of the derivative of the pseudo-Anosov we have:



In [12]:

    
s1==m*s2









    Out[12]:





True

By a Theorem of Barak Weiss and the author of this notebook, these surfaces are all periodic in the vertical direction. You can see the vertical cylinders:



In [13]:

    
s1.plot()









    Out[13]: