def has_rauzy_move(self, winner, side='right'):
r"""
Tests if the permutation is rauzy_movable on the left.
EXAMPLES:
::
sage: p = iet.Permutation('a b c','a c b',reduced=True)
sage: p.has_rauzy_move(0,'right')
True
sage: p.has_rauzy_move(0,'left')
False
sage: p.has_rauzy_move(1,'right')
True
sage: p.has_rauzy_move(1,'left')
False
::
sage: p = iet.Permutation('a b c d','c a b d',reduced=True)
sage: p.has_rauzy_move(0,'right')
False
sage: p.has_rauzy_move(0,'left')
True
sage: p.has_rauzy_move(1,'right')
False
sage: p.has_rauzy_move(1,'left')
True
"""
side = side_conversion(side)
winner = interval_conversion(winner)
return self._twin[winner][side] % len(self) != side % len(self)
def rauzy_move(self, side='right', iterations=1):
r"""
Performs a Rauzy move.
INPUT:
- ``side`` - 'left' (or 'l' or 0) or 'right' (or 'r' or 1)
- ``iterations`` - integer (default :1) the number of iteration of Rauzy
moves to perform
OUTPUT:
iet -- the Rauzy move of self
EXAMPLES::
sage: phi = QQbar((sqrt(5)-1)/2)
sage: t1 = iet.IntervalExchangeTransformation(('a b','b a'),[1,phi])
sage: t2 = t1.rauzy_move().normalize(t1.length())
sage: l2 = t2.lengths()
sage: l1 = t1.lengths()
sage: l2[0] == l1[1] and l2[1] == l1[0]
True
"""
side = side_conversion(side)
res = copy(self)
for i in range(iterations):
res = res._rauzy_move(side)
return res
def rauzy_move(self, side='right', iterations=1):
r"""
Performs a Rauzy move.
INPUT:
- ``side`` - 'left' (or 'l' or 0) or 'right' (or 'r' or 1)
- ``iterations`` - integer (default :1) the number of iteration of Rauzy
moves to perform
OUTPUT:
iet -- the Rauzy move of self
EXAMPLES::
sage: phi = QQbar((sqrt(5)-1)/2)
sage: t1 = iet.IntervalExchangeTransformation(('a b','b a'),[1,phi])
sage: t2 = t1.rauzy_move().normalize(t1.length())
sage: l2 = t2.lengths()
sage: l1 = t1.lengths()
sage: l2[0] == l1[1] and l2[1] == l1[0]
True
"""
side = side_conversion(side)
res = copy(self)
for i in range(iterations):
res = res._rauzy_move(side)
return res
def rauzy_move_relabel(self, winner, side='right'):
r"""
Returns the relabelization obtained from this move.
EXAMPLE::
sage: from surface_dynamics.all import *
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: print s_t
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: print s_b
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 surface_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 rauzy_move_relabel(self, winner, side='right'):
r"""
Returns the relabelization obtained from this move.
EXAMPLE::
sage: from surface_dynamics.all import *
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: print s_t
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: print s_b
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 surface_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 rauzy_move(self, side='right', iterations=1, data=False):
r"""
Performs a Rauzy move.
INPUT:
- ``side`` - 'left' (or 'l' or 0) or 'right' (or 'r' or 1)
- ``iterations`` - integer (default :1) the number of iteration of Rauzy
moves to perform
- ``data`` - whether to return also the paths and composition of towers
OUTPUT:
- ``iet`` -- the Rauzy move of self
- ``path`` -- (if ``data=True``) a list of 't' and 'b'
- ``towers`` -- (if ``data=True``) the towers of the Rauzy induction as a word morphism
EXAMPLES::
sage: from surface_dynamics.all import *
sage: phi = QQbar((sqrt(5)-1)/2)
sage: t1 = iet.IntervalExchangeTransformation(('a b','b a'),[1,phi])
sage: t2 = t1.rauzy_move().normalize(t1.length())
sage: l2 = t2.lengths()
sage: l1 = t1.lengths()
sage: l2[0] == l1[1] and l2[1] == l1[0]
True
sage: tt,path,sub = t1.rauzy_move(iterations=3, data=True)
sage: tt
Interval exchange transformation of [0, 0.3819660112501051?[ with
permutation
a b
b a
sage: path
['b', 't', 'b']
sage: sub
WordMorphism: a->aab, b->aabab
The substitution can also be recovered from the Rauzy diagram::
sage: p = t1.permutation()
sage: p.rauzy_diagram().path(p, *path).substitution() == sub
True
"""
if data:
towers = {a:[a] for a in self._permutation.letters()}
path = []
side = side_conversion(side)
res = copy(self)
for i in range(iterations):
res,winner,(a,b,c) = res._rauzy_move(side)
if data:
towers[a] = towers[b] + towers[c]
path.append(winner)
if data:
from sage.combinat.words.morphism import WordMorphism
return res, path, WordMorphism(towers)
else:
return res
def rauzy_move(self, side='right', iterations=1, data=False, error_on_saddles=True):
r"""
Performs a Rauzy move.
INPUT:
- ``side`` - 'left' (or 'l' or 0) or 'right' (or 'r' or 1)
- ``iterations`` - integer (default :1) the number of iteration of Rauzy
moves to perform
- ``data`` - whether to return also the paths and composition of towers
- ``error_on_saddles`` - (default: ``True``) whether to stop when a saddle
is encountered
OUTPUT:
- ``iet`` -- the Rauzy move of self
- ``path`` -- (if ``data=True``) a list of 't' and 'b'
- ``towers`` -- (if ``data=True``) the towers of the Rauzy induction as a word morphism
EXAMPLES::
sage: from surface_dynamics import *
sage: phi = QQbar((sqrt(5)-1)/2)
sage: t1 = iet.IntervalExchangeTransformation(('a b','b a'),[1,phi])
sage: t2 = t1.rauzy_move().normalize(t1.length())
sage: l2 = t2.lengths()
sage: l1 = t1.lengths()
sage: l2[0] == l1[1] and l2[1] == l1[0]
True
sage: tt,path,sub = t1.rauzy_move(iterations=3, data=True)
sage: tt
Interval exchange transformation of [0, 0.3819660112501051?[ with
permutation
a b
b a
sage: path
['b', 't', 'b']
sage: sub
WordMorphism: a->aab, b->aabab
The substitution can also be recovered from the Rauzy diagram::
sage: p = t1.permutation()
sage: p.rauzy_diagram().path(p, *path).substitution() == sub
True
An other examples involving 3 intervals::
sage: t = iet.IntervalExchangeTransformation(('a b c','c b a'),[1,1,3])
sage: t
Interval exchange transformation of [0, 5[ with permutation
a b c
c b a
sage: t1 = t.rauzy_move()
sage: t1
Interval exchange transformation of [0, 4[ with permutation
a b c
c a b
sage: t2 = t1.rauzy_move()
sage: t2
Interval exchange transformation of [0, 3[ with permutation
a b c
c b a
sage: t2.rauzy_move()
Traceback (most recent call last):
...
ValueError: saddle connection found
sage: t2.rauzy_move(error_on_saddles=False)
Interval exchange transformation of [0, 2[ with permutation
a b
a b
Degenerate cases::
sage: p = iet.Permutation('a b', 'b a')
sage: T = iet.IntervalExchangeTransformation(p, [1,1])
sage: T.rauzy_move(error_on_saddles=False)
Interval exchange transformation of [0, 1[ with permutation
a
a
"""
if data:
towers = {a:[a] for a in self._permutation.letters()}
path = []
side = side_conversion(side)
res = copy(self)
for i in range(iterations):
winner,(a,b,c) = res._rauzy_move(side, error_on_saddles=error_on_saddles)
if data:
if winner is None:
raise ValueError("does not handle degenerate situations")
towers[a] = towers[b] + towers[c]
path.append(winner)
if data:
from sage.combinat.words.morphism import WordMorphism
return res, path, WordMorphism(towers)
else:
return res
def zorich_move(self, side='right', iterations=1, data=False):
r"""
Performs a Rauzy move.
INPUT:
- ``side`` - 'left' (or 'l' or 0) or 'right' (or 'r' or 1)
- ``iterations`` - integer (default :1) the number of iteration of Rauzy
moves to perform
- ``data`` - whether to return also the path
OUTPUT:
- ``iet`` -- the Rauzy move of self
- ``path`` -- (if ``data=True``) a list of 't' and 'b'
EXAMPLES::
sage: from surface_dynamics import *
sage: p = iet.Permutation('a b c', 'c b a')
sage: T = iet.IntervalExchangeTransformation(p, [12, 35, 67])
sage: T.zorich_move()
Interval exchange transformation of [0, 55[ with permutation
a b c
c a b
sage: assert T.permutation() == p and T.lengths() == vector((12,35,67))
A self similar example in genus 2::
sage: p = iet.Permutation('a b c d', 'd a c b')
sage: R = p.rauzy_diagram()
sage: code = 'b'*4 + 't'*1 + 'b'*3 + 't'*1 + 'b'*3 + 't'*1 + 'b'*1 + 't'*1 + 'b'*4 + 't'*1 + 'b'*2 + 't'*7
sage: g = R.path(p, *code)
sage: m = g.matrix()
sage: poly = m.charpoly()
sage: l = max(poly.roots(AA, False))
sage: K.<a> = NumberField(poly, embedding=l)
sage: lengths = (m - a).right_kernel().basis()[0]
sage: T = iet.IntervalExchangeTransformation(p, lengths)
sage: T.normalize(a, inplace=True)
sage: T
Interval exchange transformation of [0, a[ with permutation
a b c d
d a c b
sage: T2, path = T.zorich_move(iterations=12, data=True)
sage: a*T2.lengths() == T.lengths()
True
sage: path == code
True
Saddle connection detection::
sage: p = iet.Permutation('a b c', 'c b a')
sage: T = iet.IntervalExchangeTransformation(p, [41, 22, 135])
sage: T.zorich_move(iterations=100)
Traceback (most recent call last):
...
ValueError: saddle connection found
sage: p = iet.Permutation('a b c d e f', 'f c b e d a')
sage: T = iet.IntervalExchangeTransformation(p, [41, 132, 22, 135, 55, 333])
sage: T.zorich_move(iterations=100)
Traceback (most recent call last):
...
ValueError: saddle connection found
"""
if data:
path = ''
side = side_conversion(side)
res = copy(self)
for i in range(iterations):
winner, m = res._zorich_move(side)
if data:
path += winner * m
if data:
return res, path
else:
return res
def rauzy_move(self,
side='right',
iterations=1,
data=False,
error_on_saddles=True):
r"""
Performs a Rauzy move.
INPUT:
- ``side`` - 'left' (or 'l' or 0) or 'right' (or 'r' or 1)
- ``iterations`` - integer (default :1) the number of iteration of Rauzy
moves to perform
- ``data`` - whether to return also the paths and composition of towers
- ``error_on_saddles`` - (default: ``True``) whether to stop when a saddle
is encountered
OUTPUT:
- ``iet`` -- the Rauzy move of self
- ``path`` -- (if ``data=True``) a list of 't' and 'b'
- ``towers`` -- (if ``data=True``) the towers of the Rauzy induction as a word morphism
EXAMPLES::
sage: from surface_dynamics.all import *
sage: phi = QQbar((sqrt(5)-1)/2)
sage: t1 = iet.IntervalExchangeTransformation(('a b','b a'),[1,phi])
sage: t2 = t1.rauzy_move().normalize(t1.length())
sage: l2 = t2.lengths()
sage: l1 = t1.lengths()
sage: l2[0] == l1[1] and l2[1] == l1[0]
True
sage: tt,path,sub = t1.rauzy_move(iterations=3, data=True)
sage: tt
Interval exchange transformation of [0, 0.3819660112501051?[ with
permutation
a b
b a
sage: path
['b', 't', 'b']
sage: sub
WordMorphism: a->aab, b->aabab
The substitution can also be recovered from the Rauzy diagram::
sage: p = t1.permutation()
sage: p.rauzy_diagram().path(p, *path).substitution() == sub
True
An other examples involving 3 intervals::
sage: t = iet.IntervalExchangeTransformation(('a b c','c b a'),[1,1,3])
sage: t
Interval exchange transformation of [0, 5[ with permutation
a b c
c b a
sage: t1 = t.rauzy_move()
sage: t1
Interval exchange transformation of [0, 4[ with permutation
a b c
c a b
sage: t2 = t1.rauzy_move()
sage: t2
Interval exchange transformation of [0, 3[ with permutation
a b c
c b a
sage: t2.rauzy_move()
Traceback (most recent call last):
...
ValueError: saddle connection found
sage: t2.rauzy_move(error_on_saddles=False)
Interval exchange transformation of [0, 2[ with permutation
a b
a b
Degenerate cases::
sage: p = iet.Permutation('a b', 'b a')
sage: T = iet.IntervalExchangeTransformation(p, [1,1])
sage: T.rauzy_move(error_on_saddles=False)
Interval exchange transformation of [0, 1[ with permutation
a
a
"""
if data:
towers = {a: [a] for a in self._permutation.letters()}
path = []
side = side_conversion(side)
res = copy(self)
for i in range(iterations):
winner, (a, b,
c) = res._rauzy_move(side,
error_on_saddles=error_on_saddles)
if data:
if winner is None:
raise ValueError("does not handle degenerate situations")
towers[a] = towers[b] + towers[c]
path.append(winner)
if data:
from sage.combinat.words.morphism import WordMorphism
return res, path, WordMorphism(towers)
else:
return res
def zorich_move(self, side='right', iterations=1, data=False):
r"""
Performs a Rauzy move.
INPUT:
- ``side`` - 'left' (or 'l' or 0) or 'right' (or 'r' or 1)
- ``iterations`` - integer (default :1) the number of iteration of Rauzy
moves to perform
- ``data`` - whether to return also the path
OUTPUT:
- ``iet`` -- the Rauzy move of self
- ``path`` -- (if ``data=True``) a list of 't' and 'b'
EXAMPLES::
sage: from surface_dynamics.all import *
sage: p = iet.Permutation('a b c', 'c b a')
sage: T = iet.IntervalExchangeTransformation(p, [12, 35, 67])
sage: T.zorich_move()
Interval exchange transformation of [0, 55[ with permutation
a b c
c a b
sage: assert T.permutation() == p and T.lengths() == vector((12,35,67))
A self similar example in genus 2::
sage: p = iet.Permutation('a b c d', 'd a c b')
sage: R = p.rauzy_diagram()
sage: code = 'b'*4 + 't'*1 + 'b'*3 + 't'*1 + 'b'*3 + 't'*1 + 'b'*1 + 't'*1 + 'b'*4 + 't'*1 + 'b'*2 + 't'*7
sage: g = R.path(p, *code)
sage: m = g.matrix()
sage: poly = m.charpoly()
sage: l = max(poly.roots(AA, False))
sage: K.<a> = NumberField(poly, embedding=l)
sage: lengths = (m - a).right_kernel().basis()[0]
sage: T = iet.IntervalExchangeTransformation(p, lengths)
sage: T.normalize(a, inplace=True)
sage: T
Interval exchange transformation of [0, a[ with permutation
a b c d
d a c b
sage: T2, path = T.zorich_move(iterations=12, data=True)
sage: a*T2.lengths() == T.lengths()
True
sage: path == code
True
Saddle connection detection::
sage: p = iet.Permutation('a b c', 'c b a')
sage: T = iet.IntervalExchangeTransformation(p, [41, 22, 135])
sage: T.zorich_move(iterations=100)
Traceback (most recent call last):
...
ValueError: saddle connection found
sage: p = iet.Permutation('a b c d e f', 'f c b e d a')
sage: T = iet.IntervalExchangeTransformation(p, [41, 132, 22, 135, 55, 333])
sage: T.zorich_move(iterations=100)
Traceback (most recent call last):
...
ValueError: saddle connection found
"""
if data:
path = ''
side = side_conversion(side)
res = copy(self)
for i in range(iterations):
winner, m = res._zorich_move(side)
if data:
path += winner * m
if data:
return res, path
else:
return res