def __init__(self, mapping, inv_mapping, l_edges, r_edges, s2_edges,
s3_edges):
N = len(s2_edges)
ss2 = [None] * len(s2_edges)
ss3 = [None] * len(s3_edges)
ll = [None] * len(l_edges)
rr = [None] * len(r_edges)
for i in range(N):
ss2[s2_edges[i]] = i
ss3[s3_edges[i]] = i
ll[l_edges[i]] = i
rr[r_edges[i]] = i
if mapping[0].is_orientation_cover():
self._veech_group = EvenArithmeticSubgroup_Permutation(
ss2, ss3, ll, rr)
else:
self._veech_group = OddArithmeticSubgroup_Permutation(
ss2, ss3, ll, rr)
self._l_edges = l_edges
self._li_edges = ll
self._r_edges = r_edges
self._ri_edges = rr
self._s2_edges = s2_edges
self._s2i_edges = ss2
self._s3_edges = s3_edges
self._s3i_edges = ss3
self._mapping = mapping
self._inv_mapping = inv_mapping
def __init__(self, mapping, inv_mapping, l_edges, r_edges, s2_edges, s3_edges):
N = len(s2_edges)
ss2 = [None] * len(s2_edges)
ss3 = [None] * len(s3_edges)
ll = [None] * len(l_edges)
rr = [None] * len(r_edges)
for i in xrange(N):
ss2[s2_edges[i]] = i
ss3[s3_edges[i]] = i
ll[l_edges[i]] = i
rr[r_edges[i]] = i
if mapping[0].is_orientation_cover():
self._veech_group = EvenArithmeticSubgroup_Permutation(ss2,ss3,ll,rr)
else:
self._veech_group = OddArithmeticSubgroup_Permutation(ss2,ss3,ll,rr)
self._l_edges = l_edges
self._li_edges = ll
self._r_edges = r_edges
self._ri_edges = rr
self._s2_edges = s2_edges
self._s2i_edges = ss2
self._s3_edges = s3_edges
self._s3i_edges = ss3
self._mapping = mapping
self._inv_mapping = inv_mapping
def as_permutation_group(self):
r"""
Return self as an arithmetic subgroup defined in terms of the
permutation action of `SL(2,\ZZ)` on its right cosets.
This method uses Todd-Coxeter enumeration (via the method
:meth:`~todd_coxeter`) which can be extremely slow for arithmetic
subgroups with relatively large index in `SL(2,\ZZ)`.
EXAMPLES::
sage: G = Gamma(3)
sage: P = G.as_permutation_group(); P
Arithmetic subgroup of index 24
sage: G.ncusps() == P.ncusps()
True
sage: G.nu2() == P.nu2()
True
sage: G.nu3() == P.nu3()
True
sage: G.an_element() in P
True
sage: P.an_element() in G
True
"""
_, _, l_edges, s2_edges = self.todd_coxeter()
n = len(l_edges)
s3_edges = [None] * n
r_edges = [None] * n
for i in xrange(n):
ii = s2_edges[l_edges[i]]
s3_edges[ii] = i
r_edges[ii] = s2_edges[i]
if self.is_even():
from sage.modular.arithgroup.arithgroup_perm import EvenArithmeticSubgroup_Permutation
g = EvenArithmeticSubgroup_Permutation(S2=s2_edges,
S3=s3_edges,
L=l_edges,
R=r_edges)
else:
from sage.modular.arithgroup.arithgroup_perm import OddArithmeticSubgroup_Permutation
g = OddArithmeticSubgroup_Permutation(S2=s2_edges,
S3=s3_edges,
L=l_edges,
R=r_edges)
g.relabel()
return g
class TeichmuellerCurveOfOrigami_class(TeichmuellerCurve):
def __init__(self, mapping, inv_mapping, l_edges, r_edges, s2_edges,
s3_edges):
N = len(s2_edges)
ss2 = [None] * len(s2_edges)
ss3 = [None] * len(s3_edges)
ll = [None] * len(l_edges)
rr = [None] * len(r_edges)
for i in range(N):
ss2[s2_edges[i]] = i
ss3[s3_edges[i]] = i
ll[l_edges[i]] = i
rr[r_edges[i]] = i
if mapping[0].is_orientation_cover():
self._veech_group = EvenArithmeticSubgroup_Permutation(
ss2, ss3, ll, rr)
else:
self._veech_group = OddArithmeticSubgroup_Permutation(
ss2, ss3, ll, rr)
self._l_edges = l_edges
self._li_edges = ll
self._r_edges = r_edges
self._ri_edges = rr
self._s2_edges = s2_edges
self._s2i_edges = ss2
self._s3_edges = s3_edges
self._s3i_edges = ss3
self._mapping = mapping
self._inv_mapping = inv_mapping
def __reduce__(self):
return (TeichmuellerCurveOfOrigami_class,
(self._mapping, self._inv_mapping, self._l_edges,
self._r_edges, self._s2_edges, self._s3_edges))
def _repr_(self):
r"""
string representation of self
"""
return "Teichmueller curve of the origami\n%s" % self.origami()
def stratum(self):
r"""
Returns the stratum of the Teichmueller curve
EXAMPLES::
sage: from surface_dynamics import *
sage: o = Origami('(1,2)','(1,3)')
sage: t = o.teichmueller_curve()
sage: t.stratum()
H_2(2)
"""
return self[0].stratum()
def origami(self):
r"""
Returns an origami in this Teichmueller curve.
Beware that the labels of the initial origami may have changed.
EXAMPLES::
sage: from surface_dynamics import *
sage: o = Origami('(1,2)','(1,3)')
sage: t = o.teichmueller_curve()
sage: t.origami()
(1)(2,3)
(1,2)(3)
"""
return self._mapping[0]
an_element = origami
def orbit_graph(self,
s2_edges=True,
s3_edges=True,
l_edges=False,
r_edges=False,
vertex_labels=True):
r"""
Return the graph of action of PSL on this origami
INPUT:
- ``return_map`` - return the list of origamis in the orbit
"""
G = self._veech_group.coset_graph(s2_edges=s2_edges,
s3_edges=s3_edges,
l_edges=l_edges,
r_edges=r_edges)
if vertex_labels:
G.relabel(dict(enumerate(self._mapping)))
return G
def __getitem__(self, i):
assert (isinstance(i, (int, Integer)))
if i < 0 or i >= len(self._mapping):
raise IndexError
return self._mapping[i]
def veech_group(self):
r"""
Return the veech group of the Teichmueller curve
"""
return self._veech_group
def plot_graph(self):
r"""
Plot the graph of the veech group.
The graph corresponds to the action of the generators on the cosets
determined by the Veech group in PSL(2,ZZ).
"""
return self.orbit_graph().plot()
# def plot_fundamental_domain(self):
# from sage.geometry.hyperbolic_geometry import HH
# from math import cos,sin,pi
# from sage.matrix.constructor import matrix
# from sage.all import CC
# S=matrix([[0,-1],[1,0]])
# T=matrix([[1,1],[0,1]])
# g3=S*T
# tree,gen,lifts=self._graph.spanning_tree(gq=g3)
# pol = HH.polygon([Infinity,CC(0,1),CC(0,0),CC(-cos(pi/3),sin(pi/3))])
# res = pol.plot(face_color='blue')
# for m in lifts[1:]:
# res += (m*pol).plot(face_color='green',face_alpha=0.3)
# return res
def cusp_representative_iterator(self):
r"""
Iterator over the cusp of self.
Each term is a couple ``(o,w)`` where ``o`` is a representative of the
cusp (an origami) and ``w`` is the width of the cusp (an integer).
"""
n = len(self._mapping)
seen = [True] * n
l = self._l_edges
for i in range(n):
if seen[i]:
k = 1
seen[i] = False
j = l[i]
while j != i:
seen[j] = False
j = l[j]
k += 1
yield (self._mapping[i], k)
def cusp_representatives(self):
return list(self.cusp_representative_iterator())
def sum_of_lyapunov_exponents(self):
r"""
Returns the sum of Lyapunov exponents for this origami
EXAMPLES::
sage: from surface_dynamics import *
Let us consider few examples in H(2) for which the sum is independent of
the origami::
sage: o = Origami('(1,2)','(1,3)')
sage: o.stratum()
H_2(2)
sage: o.sum_of_lyapunov_exponents()
4/3
sage: o = Origami('(1,2,3)(4,5,6)','(1,5,7)(2,6)(3,4)')
sage: o.stratum()
H_2(2)
sage: o.sum_of_lyapunov_exponents()
4/3
This is true as well for the stratum H(1,1)::
sage: o = Origami('(1,2)','(1,3)(2,4)')
sage: o.stratum()
H_2(1^2)
sage: o.sum_of_lyapunov_exponents()
3/2
sage: o = Origami('(1,2,3,4,5,6,7)','(2,6)(3,7)')
sage: o.stratum()
H_2(1^2)
sage: o.sum_of_lyapunov_exponents()
3/2
ALGORITHM:
Kontsevich-Zorich formula
"""
K = Integer(1) / Integer(12) * sum(m * (m + 2) / (m + 1)
for m in self.stratum().zeros())
KK = 0
for o in self._mapping:
KK += sum(w / h for (h, w) in o.widths_and_heights())
return K + Integer(1) / Integer(len(self._mapping)) * KK
# TODO: mysterious signs and interversion problems...
def simplicial_action_generators(self):
r"""
Return action of generators of the Veech group on homology
"""
from sage.matrix.constructor import matrix, identity_matrix
tree, reps, word_reps, gens = self._veech_group._spanning_tree_verrill(
on_right=False)
n = self[0].nb_squares()
l_mat = identity_matrix(2 * n)
s_mat = matrix(2 * n)
s_mat[:n, n:] = -identity_matrix(n)
reps = [None] * len(tree)
reps_o = [None] * len(tree)
reps[0] = identity_matrix(2 * n)
reps_o[0] = self.origami()
waiting = [0]
while waiting:
x = waiting.pop()
for (e0, e1, label) in tree.outgoing_edges(x):
waiting.append(e1)
o = reps_o[e0]
if label == 's':
m = copy(s_mat)
m[n:, :n] = o.u().matrix().transpose()
reps[e1] = m * reps[e0]
reps_o[e1] = o.S_action()
elif label == 'l':
o = o.L_action()
m = copy(l_mat)
m[:n, n:] = (~o.u()).matrix().transpose()
reps[e1] = m * reps[e0]
reps_o[e1] = o
elif label == 'r':
o = o.R_action()
m = copy(l_mat)
m[n:, :n] = (~o.r()).matrix().transpose()
reps[e1] = m * reps[e0]
reps_o[e1] = o
else:
raise ValueError("does know how to do only with slr")
m_gens = []
for (e0, e1, label) in gens:
o = reps_o[e0]
if label == 's':
oo = o.S_action()
m = copy(s_mat)
m[n:, :n] = o.u().matrix().transpose()
elif label == 'l':
oo = o.L_action()
m = copy(l_mat)
m[:n, n:] = (~oo.u()).matrix().transpose()
elif label == 'r':
oo = o.R_action()
m = copy(l_mat)
m[:n, n:] = (~oo.r()).matrix().transpose()
else:
raise ValueError("does know how to do only with slr")
ooo = reps_o[e1]
ot, oo_renum = oo.standard_form(True)
os, ooo_renum = ooo.standard_form(True)
assert (ot == os)
m_ren = (oo_renum * ~ooo_renum).matrix().transpose()
m_s = matrix(2 * n)
m_s[:n, :n] = m_ren
m_s[n:, n:] = m_ren
m_gens.append(~reps[e1] * m_s * m * reps[e0])
return tree, reps, reps_o, gens, m_gens, word_reps
class TeichmuellerCurveOfOrigami_class(TeichmuellerCurve):
def __init__(self, mapping, inv_mapping, l_edges, r_edges, s2_edges, s3_edges):
N = len(s2_edges)
ss2 = [None] * len(s2_edges)
ss3 = [None] * len(s3_edges)
ll = [None] * len(l_edges)
rr = [None] * len(r_edges)
for i in xrange(N):
ss2[s2_edges[i]] = i
ss3[s3_edges[i]] = i
ll[l_edges[i]] = i
rr[r_edges[i]] = i
if mapping[0].is_orientation_cover():
self._veech_group = EvenArithmeticSubgroup_Permutation(ss2,ss3,ll,rr)
else:
self._veech_group = OddArithmeticSubgroup_Permutation(ss2,ss3,ll,rr)
self._l_edges = l_edges
self._li_edges = ll
self._r_edges = r_edges
self._ri_edges = rr
self._s2_edges = s2_edges
self._s2i_edges = ss2
self._s3_edges = s3_edges
self._s3i_edges = ss3
self._mapping = mapping
self._inv_mapping = inv_mapping
def __reduce__(self):
return (TeichmuellerCurveOfOrigami_class,
(self._mapping, self._inv_mapping,
self._l_edges, self._r_edges,
self._s2_edges, self._s3_edges))
def _repr_(self):
r"""
string representation of self
"""
return "Teichmueller curve of the origami\n%s" %self.origami()
def stratum(self):
r"""
Returns the stratum of the Teichmueller curve
EXAMPLES::
sage: from surface_dynamics.all import *
sage: o = Origami('(1,2)','(1,3)')
sage: t = o.teichmueller_curve()
sage: t.stratum()
H_2(2)
"""
return self[0].stratum()
def origami(self):
r"""
Returns an origmami in this teichmueller curve.
Beware that the labels of the initial origami may have changed.
EXAMPLES::
sage: from surface_dynamics.all import *
sage: o = Origami('(1,2)','(1,3)')
sage: t = o.teichmueller_curve()
sage: t.origami()
(1)(2,3)
(1,2)(3)
"""
return self._mapping[0]
an_element = origami
def orbit_graph(self,
s2_edges=True, s3_edges=True, l_edges=False, r_edges=False,
vertex_labels=True):
r"""
Return the graph of action of PSL on this origami
INPUT:
- ``return_map`` - return the list of origamis in the orbit
"""
G = self._veech_group.coset_graph(
s2_edges=s2_edges,
s3_edges=s3_edges,
l_edges=l_edges,
r_edges=r_edges)
if vertex_labels:
G.relabel(dict(enumerate(self._mapping)))
return G
def __getitem__(self,i):
assert(isinstance(i,(int,Integer)))
if i < 0 or i >= len(self._mapping):
raise IndexError
return self._mapping[i]
def veech_group(self):
r"""
Return the veech group of the Teichmueller curve
"""
return self._veech_group
def plot_graph(self):
r"""
Plot the graph of the veech group.
The graph corresponds to the action of the generators on the cosets
determined by the Veech group in PSL(2,ZZ).
"""
return self.orbit_graph.plot()
# def plot_fundamental_domain(self):
# from sage.geometry.hyperbolic_geometry import HH
# from math import cos,sin,pi
# from sage.matrix.constructor import matrix
# from sage.all import CC
# S=matrix([[0,-1],[1,0]])
# T=matrix([[1,1],[0,1]])
# g3=S*T
# tree,gen,lifts=self._graph.spanning_tree(gq=g3)
# pol = HH.polygon([Infinity,CC(0,1),CC(0,0),CC(-cos(pi/3),sin(pi/3))])
# res = pol.plot(face_color='blue')
# for m in lifts[1:]:
# res += (m*pol).plot(face_color='green',face_alpha=0.3)
# return res
def cusp_representative_iterator(self):
r"""
Iterator over the cusp of self.
Each term is a couple ``(o,w)`` where ``o`` is a representative of the
cusp (an origami) and ``w`` is the width of the cusp (an integer).
"""
cusps = []
n = len(self._mapping)
seen = [True]*n
l = self._l_edges
for i in xrange(n):
if seen[i]:
k=1
seen[i] = False
j = l[i]
while j != i:
seen[j] = False
j = l[j]
k += 1
yield (self._mapping[i],k)
def cusp_representatives(self):
return list(self.cusp_representative_iterator())
def sum_of_lyapunov_exponents(self):
r"""
Returns the sum of Lyapunov exponents for this origami
EXAMPLES::
sage: from surface_dynamics.all import *
Let us consider few examples in H(2) for which the sum is independant of
the origami::
sage: o = Origami('(1,2)','(1,3)')
sage: o.stratum()
H_2(2)
sage: o.sum_of_lyapunov_exponents()
4/3
sage: o = Origami('(1,2,3)(4,5,6)','(1,5,7)(2,6)(3,4)')
sage: o.stratum()
H_2(2)
sage: o.sum_of_lyapunov_exponents()
4/3
This is true as well for the stratum H(1,1)::
sage: o = Origami('(1,2)','(1,3)(2,4)')
sage: o.stratum()
H_2(1^2)
sage: o.sum_of_lyapunov_exponents()
3/2
sage: o = Origami('(1,2,3,4,5,6,7)','(2,6)(3,7)')
sage: o.stratum()
H_2(1^2)
sage: o.sum_of_lyapunov_exponents()
3/2
ALGORITHM:
Kontsevich-Zorich formula
"""
K = Integer(1)/Integer(12) * sum(m*(m+2)/(m+1) for m in self.stratum().zeros())
KK = 0
for o in self._mapping:
KK += sum(w/h for (h,w) in o.widths_and_heights())
return K + Integer(1)/Integer(len(self._mapping)) * KK
#TODO: mysterious signs and interversion problems...
def simplicial_action_generators(self):
r"""
Return action of generators of the Veech group on homolgy
"""
from sage.matrix.constructor import matrix, identity_matrix
tree,reps,word_reps,gens = self._veech_group._spanning_tree_verrill(on_right=False)
n = self[0].nb_squares()
l_mat = identity_matrix(2*n)
s_mat = matrix(2*n)
s_mat[:n,n:] = -identity_matrix(n)
reps = [None]*len(tree)
reps_o = [None]*len(tree)
reps[0] = identity_matrix(2*n)
reps_o[0] = self.origami()
waiting = [0]
while waiting:
x = waiting.pop()
for (e0,e1,label) in tree.outgoing_edges(x):
waiting.append(e1)
o = reps_o[e0]
if label == 's':
m = copy(s_mat)
m[n:,:n] = o.y().matrix().transpose()
reps[e1] = m * reps[e0]
reps_o[e1] = o.S_action()
elif label == 'l':
o = o.L_action()
m = copy(l_mat)
m[:n,n:] = (~o.y()).matrix().transpose()
reps[e1] = m * reps[e0]
reps_o[e1] = o
elif label == 'r':
o = o.R_action()
m = copy(l_mat)
m[n:,:n] = (~o.x()).matrix().transpose()
reps[e1] = m * reps[e0]
reps_o[e1] = o
else:
raise ValueError, "does know how to do only with slr"
m_gens = []
for (e0,e1,label) in gens:
o = reps_o[e0]
if label == 's':
oo = o.S_action()
m = copy(s_mat)
m[n:,:n] = o.y().matrix().transpose()
elif label == 'l':
oo = o.L_action()
m = copy(l_mat)
m[:n,n:] = (~oo.y()).matrix().transpose()
elif label == 'r':
oo = o.R_action()
m = copy(l_mat)
m[:n,n:] = (~oo.x()).matrix().transpose()
else:
raise ValueError, "does know how to do only with slr"
ooo = reps_o[e1]
ot, oo_renum = oo.standard_form(True)
os, ooo_renum = ooo.standard_form(True)
assert(ot == os)
m_ren = (oo_renum * ~ooo_renum).matrix().transpose()
m_s = matrix(2*n)
m_s[:n,:n] = m_ren;
m_s[n:,n:] = m_ren
m_gens.append(~reps[e1] * m_s * m * reps[e0])
return tree,reps,reps_o,gens,m_gens,word_reps
