def rauzy_move_relabel(self, winner, side='right'):
r"""
Returns the relabelization obtained from this move.
EXAMPLE::
sage: p = iet.Permutation('a b c d','d c b a')
sage: q = p.reduced()
sage: p_t = p.rauzy_move('t')
sage: q_t = q.rauzy_move('t')
sage: s_t = q.rauzy_move_relabel('t')
sage: s_t
WordMorphism: a->a, b->b, c->c, d->d
sage: map(s_t, p_t[0]) == map(Word, q_t[0])
True
sage: map(s_t, p_t[1]) == map(Word, q_t[1])
True
sage: p_b = p.rauzy_move('b')
sage: q_b = q.rauzy_move('b')
sage: s_b = q.rauzy_move_relabel('b')
sage: s_b
WordMorphism: a->a, b->d, c->b, d->c
sage: map(s_b, q_b[0]) == map(Word, p_b[0])
True
sage: map(s_b, q_b[1]) == map(Word, p_b[1])
True
"""
from sage.dynamics.interval_exchanges.labelled import LabelledPermutationIET
from sage.combinat.words.morphism import WordMorphism
winner = interval_conversion(winner)
side = side_conversion(side)
p = LabelledPermutationIET(self.list())
l0_q = p.rauzy_move(winner, side).list()[0]
d = dict([(self._alphabet[i],l0_q[i]) for i in range(len(self))])
return WordMorphism(d)
def representative(self, reduced=True, alphabet=None):
"""
Returns the Zorich representative of this connected component.
Zorich constructs explicitely interval exchange
transformations for each stratum in [Zor08]_.
EXAMPLES:
::
sage: a = AbelianStratum(6).connected_components()[1]
sage: print a.representative(alphabet=range(8))
0 1 2 3 4 5 6 7
3 2 5 4 7 6 1 0
::
sage: a = AbelianStratum(4,4).connected_components()[1]
sage: print a.representative(alphabet=range(11))
0 1 2 3 4 5 6 7 8 9 10
3 2 5 4 6 8 7 10 9 1 0
"""
zeroes = [x//2 for x in self._parent._zeroes if x > 0]
n = self._parent._zeroes.count(0)
g = self._parent._genus
l0 = range(3*g-2)
l1 = [3, 2]
for k in range(4, 3*g-4, 3):
l1 += [k, k+2, k+1]
l1 += [1, 0]
k = 4
for d in zeroes:
for i in range(d-1):
del l0[l0.index(k)]
del l1[l1.index(k)]
k += 3
k += 3
# marked points
if n != 0:
interval = range(3*g-2, 3*g-2+n)
if self._parent._zeroes[0] == 0:
k = l0.index(3)
l0[k:k] = interval
l1[-1:-1] = interval
else:
l0[1:1] = interval
l1.extend(interval)
if self._parent._marked_separatrix == 'in':
l0.reverse()
l1.reverse()
if reduced:
from sage.dynamics.interval_exchanges.reduced import ReducedPermutationIET
return ReducedPermutationIET([l0, l1], alphabet=alphabet)
else:
from sage.dynamics.interval_exchanges.labelled import LabelledPermutationIET
return LabelledPermutationIET([l0, l1], alphabet=alphabet)
def representative(self, reduced=True, alphabet=None):
r"""
Returns the Zorich representative of this connected component.
Zorich constructs explicitely interval exchange
transformations for each stratum in [Zor08]_.
EXAMPLES:
::
sage: c = AbelianStratum(6).connected_components()[2]
sage: c
H_even(6)
sage: p = c.representative(alphabet=range(8))
sage: p
0 1 2 3 4 5 6 7
5 4 3 2 7 6 1 0
sage: p.connected_component()
H_even(6)
::
sage: c = AbelianStratum(4,4).connected_components()[2]
sage: c
H_even(4, 4)
sage: p = c.representative(alphabet=range(11))
sage: p
0 1 2 3 4 5 6 7 8 9 10
5 4 3 2 6 8 7 10 9 1 0
sage: p.connected_component()
H_even(4, 4)
"""
zeroes = [x for x in self._parent._zeroes if x > 0]
n = self._parent._zeroes.count(0)
g = self._parent._genus
l0 = range(3*g-2)
l1 = [6, 5, 4, 3, 2, 7, 9, 8]
for k in range(10, 3*g-4, 3):
l1 += [k, k+2, k+1]
l1 += [1, 0]
k = 4
for d in zeroes:
for i in range(d/2-1):
del l0[l0.index(k)]
del l1[l1.index(k)]
k += 3
k += 3
# if there are marked points we transform 0 in [3g-2, 3g-3, ...]
if n != 0:
interval = range(3*g-2, 3*g - 2 + n)
if self._parent._zeroes[0] == 0:
k = l0.index(6)
l0[k:k] = interval
l1[-1:-1] = interval
else:
l0[1:1] = interval
l1.extend(interval)
if self._parent._marked_separatrix == 'in':
l0.reverse()
l1.reverse()
if reduced:
from sage.dynamics.interval_exchanges.reduced import ReducedPermutationIET
return ReducedPermutationIET([l0, l1], alphabet=alphabet)
else:
from sage.dynamics.interval_exchanges.labelled import LabelledPermutationIET
return LabelledPermutationIET([l0, l1], alphabet=alphabet)
def representative(self, reduced=True, alphabet=None):
r"""
Returns the Zorich representative of this connected component.
Zorich constructs explicitely interval exchange
transformations for each stratum in [Zor08]_.
INPUT:
- ``reduced`` - boolean (defaut: ``True``): whether you obtain
a reduced or labelled permutation
- ``alphabet`` - alphabet or ``None`` (defaut: ``None``):
whether you want to specify an alphabet for your
representative
EXAMPLES:
::
sage: c = AbelianStratum(0).connected_components()[0]
sage: c
H_hyp(0)
sage: p = c.representative(alphabet="01")
sage: p
0 1
1 0
sage: p.connected_component()
H_hyp(0)
::
sage: c = AbelianStratum(0,0).connected_components()[0]
sage: c
H_hyp(0, 0)
sage: p = c.representative(alphabet="abc")
sage: p
a b c
c b a
sage: p.connected_component()
H_hyp(0, 0)
::
sage: c = AbelianStratum(2).connected_components()[0]
sage: c
H_hyp(2)
sage: p = c.representative(alphabet="ABCD")
sage: p
A B C D
D C B A
sage: p.connected_component()
H_hyp(2)
::
sage: c = AbelianStratum(1,1).connected_components()[0]
sage: c
H_hyp(1, 1)
sage: p = c.representative(alphabet="01234")
sage: p
0 1 2 3 4
4 3 2 1 0
sage: p.connected_component()
H_hyp(1, 1)
"""
g = self._parent._genus
n = self._parent._zeroes.count(0)
m = len(self._parent._zeroes) - n
if m == 0: # on the torus
if n == 1:
l0 = [0, 1]
l1 = [1, 0]
elif n == 2:
l0 = [0, 1, 2]
l1 = [2, 1, 0]
else:
l0 = range(1, n+2)
l1 = [n+1] + range(1, n+1)
elif m == 1: # H(2g-2,0^n) or H(0,2g-2,0^(n-1))
l0 = range(1, 2*g+1)
l1 = range(2*g, 0, -1)
interval = range(2*g+1, 2*g+n+1)
if self._parent._zeroes[0] == 0:
l0[-1:-1] = interval
l1[-1:-1] = interval
else:
l0[1:1] = interval
l1[1:1] = interval
else: # H(g-1,g-1,0^n) or H(0,g-1,g-1,0^(n-1))
l0 = range(1, 2*g+2)
l1 = range(2*g+1, 0, -1)
interval = range(2*g+2, 2*g+n+2)
if self._parent._zeroes[0] == 0:
l0[-1:-1] = interval
l1[-1:-1] = interval
else:
l0[1:1] = interval
l1[1:1] = interval
if self._parent._marked_separatrix == 'in':
l0.reverse()
l1.reverse()
if reduced:
from sage.dynamics.interval_exchanges.reduced import ReducedPermutationIET
return ReducedPermutationIET([l0, l1], alphabet=alphabet)
else:
from sage.dynamics.interval_exchanges.labelled import LabelledPermutationIET
return LabelledPermutationIET([l0, l1], alphabet=alphabet)
def representative(self, reduced=True, alphabet=None):
r"""
Returns the Zorich representative of this connected component.
Zorich constructs explicitely interval exchange
transformations for each stratum in [Zor08]_.
INPUT:
- ``reduced`` - boolean (default: ``True``): whether you
obtain a reduced or labelled permutation
- ``alphabet`` - an alphabet or ``None``: whether you want to
specify an alphabet for your permutation
OUTPUT:
permutation -- a permutation which lives in this component
EXAMPLES:
::
sage: c = AbelianStratum(1,1,1,1).connected_components()[0]
sage: print c
H_c(1, 1, 1, 1)
sage: p = c.representative(alphabet=range(9))
sage: print p
0 1 2 3 4 5 6 7 8
4 3 2 5 8 7 6 1 0
sage: p.connected_component()
H_c(1, 1, 1, 1)
"""
g = self._parent._genus
zeroes = [x for x in self._parent._zeroes if x > 0]
n = self._parent._zeroes.count(0)
l0 = range(0, 4*g-3)
l1 = [4, 3, 2]
for k in range(5, 4*g-6, 4):
l1 += [k, k+3, k+2, k+1]
l1 += [1, 0]
k = 3
for d in zeroes:
for i in range(d-1):
del l0[l0.index(k)]
del l1[l1.index(k)]
k += 2
k += 2
if n != 0:
interval = range(4*g-3, 4*g-3+n)
if self._parent._zeroes[0] == 0:
k = l0.index(4)
l0[k:k] = interval
l1[-1:-1] = interval
else:
l0[1:1] = interval
l1.extend(interval)
if self._parent._marked_separatrix == 'in':
l0.reverse()
l1.reverse()
if reduced:
from sage.dynamics.interval_exchanges.reduced import ReducedPermutationIET
return ReducedPermutationIET([l0, l1], alphabet=alphabet)
else:
from sage.dynamics.interval_exchanges.labelled import LabelledPermutationIET
return LabelledPermutationIET([l0, l1], alphabet=alphabet)
