r"""
Class factories for Interval exchange transformations.
This library is designed for the usage and manipulation of interval
exchange transformations and linear involutions. It defines specialized
types of permutation (constructed using :meth:`iet.Permutation`) some
associated graph (constructed using :meth:`iet.RauzyGraph`) and some maps
of intervals (constructed using :meth:`iet.IntervalExchangeTransformation`).
EXAMPLES:
Creation of an interval exchange transformation::
sage: T = iet.IntervalExchangeTransformation(('a b','b a'),(sqrt(2),1))
sage: print T
Interval exchange transformation of [0, sqrt(2) + 1[ with permutation
a b
b a
It can also be initialized using permutation (group theoritic ones)::
sage: p = Permutation([3,2,1])
sage: T = iet.IntervalExchangeTransformation(p, [1/3,2/3,1])
sage: print T
Interval exchange transformation of [0, 2[ with permutation
1 2 3
3 2 1
For the manipulation of permutations of iet, there are special types provided
by this module. All of them can be constructed using the constructor
iet.Permutation. For the creation of labelled permutations of interval exchange
transformation::
sage: p1 = iet.Permutation('a b c', 'c b a')
sage: print p1
a b c
c b a
They can be used for initialization of an iet::
sage: p = iet.Permutation('a b','b a')
sage: T = iet.IntervalExchangeTransformation(p, [1,sqrt(2)])
sage: print T
Interval exchange transformation of [0, sqrt(2) + 1[ with permutation
a b
b a
You can also, create labelled permutations of linear involutions::
sage: p = iet.GeneralizedPermutation('a a b', 'b c c')
sage: print p
a a b
b c c
Sometimes it's more easy to deal with reduced permutations::
sage: p = iet.Permutation('a b c', 'c b a', reduced = True)
sage: print p
a b c
c b a
Permutations with flips::
sage: p1 = iet.Permutation('a b c', 'c b a', flips = ['a','c'])
sage: print p1
-a b -c
-c b -a
Creation of Rauzy diagrams::
sage: r = iet.RauzyDiagram('a b c', 'c b a')
Reduced Rauzy diagrams are constructed using the same arguments than for
permutations::
sage: r = iet.RauzyDiagram('a b b','c c a')
sage: r_red = iet.RauzyDiagram('a b b','c c a',reduced=True)
sage: r.cardinality()
12
sage: r_red.cardinality()
4
By defaut, Rauzy diagram are generated by induction on the right. You can use
several options to enlarge (or restrict) the diagram (try help(iet.RauzyDiagram) for
more precisions)::
sage: r1 = iet.RauzyDiagram('a b c','c b a',right_induction=True)
sage: r2 = iet.RauzyDiagram('a b c','c b a',left_right_inversion=True)
You can consider self similar iet using path in Rauzy diagrams and eigenvectors
of the corresponding matrix::
sage: p = iet.Permutation("a b c d", "d c b a")
sage: d = p.rauzy_diagram()
sage: g = d.path(p, 't', 't', 'b', 't', 'b', 'b', 't', 'b')
sage: g
Path of length 8 in a Rauzy diagram
sage: g.is_loop()
True
sage: g.is_full()
True
sage: m = g.matrix()
sage: v = m.eigenvectors_right()[-1][1][0]
sage: T1 = iet.IntervalExchangeTransformation(p, v)
sage: T2 = T1.rauzy_move(iterations=8)
sage: T1.normalize(1) == T2.normalize(1)
True
REFERENCES:
.. [BL08] Corentin Boissy and Erwan Lanneau, "Dynamics and geometry of the
Rauzy-Veech induction for quadratic differentials" (arxiv:0710.5614) to appear
in Ergodic Theory and Dynamical Systems
.. [DN90] Claude Danthony and Arnaldo Nogueira "Measured foliations on
nonorientable surfaces", Annales scientifiques de l'Ecole Normale
Superieure, Ser. 4, 23, no. 3 (1990) p 469-494
.. [N85] Arnaldo Nogueira, "Almost all Interval Exchange Transformations with
Flips are Nonergodic" (Ergod. Th. & Dyn. Systems, Vol 5., (1985), 257-271
.. [R79] Gerard Rauzy, "Echanges d'intervalles et transformations induites",
Acta Arith. 34, no. 3, 203-212, 1980
.. [V78] William Veech, "Interval exchange transformations", J. Analyse Math.
33, 222-272
.. [Z] Anton Zorich, "Generalized Permutation software"
(http://perso.univ-rennes1.fr/anton.zorich)
AUTHORS:
- Vincent Delecroix (2009-09-29): initial version
"""
def _two_lists(a):
r"""
Try to return the input as a list of two lists
INPUT:
- ``a`` - either a string, one or two lists, one or two tuples
OUTPUT:
-- two lists
TESTS:
::
sage: from sage.combinat.iet.constructors import _two_lists
sage: _two_lists(('a1 a2','b1 b2'))
[['a1', 'a2'], ['b1', 'b2']]
sage: _two_lists('a1 a2\nb1 b2')
[['a1', 'a2'], ['b1', 'b2']]
sage: _two_lists(['a b','c'])
[['a', 'b'], ['c']]
..The ValueError and TypeError can be raised if it fails::
sage: _two_lists('a b')
Traceback (most recent call last):
...
ValueError: your chain must contain two lines
sage: _two_lists('a b\nc d\ne f')
Traceback (most recent call last):
...
ValueError: your chain must contain two lines
sage: _two_lists(1)
Traceback (most recent call last):
...
TypeError: argument not accepted
"""
from sage.combinat.permutation import Permutation_class
res = [None,None]
if isinstance(a,str):
a = a.split('\n')
if len(a) != 2:
raise ValueError, "your chain must contain two lines"
else :
res[0] = a[0].split()
res[1] = a[1].split()
elif isinstance(a, Permutation_class):
res[0] = range(1,len(a)+1)
res[1] = [a[i] for i in range(len(a))]
elif not hasattr(a,'__len__'):
raise TypeError, "argument not accepted"
elif len(a) == 0 or len(a) > 2:
raise ValueError, "your argument can not be split in two parts"
elif len(a) == 1:
a = a[0]
if isinstance(a, Permutation_class):
res[0] = range(1,len(a)+1)
res[1] = [a[i] for i in range(len(a))]
elif isinstance(a, (list,tuple)):
if (len(a) != 2):
raise ValueError, "your list must contain two objects"
for i in range(2):
if isinstance(a[i], str):
res[i] = a[i].split()
else:
res[i] = list(a[i])
else :
raise TypeError, "argument not accepted"
else :
for i in range(2):
if isinstance(a[i], str):
res[i] = a[i].split()
else:
res[i] = list(a[i])
return res
def Permutation(*args,**kargs):
r"""
Returns a permutation of an interval exchange transformation.
Those permutations are the combinatoric part of an interval exchange
transformation (IET). The combinatorial study of those objects starts with
Gerard Rauzy [R79]_ and William Veech [V78]_.
The combinatoric part of interval exchange transformation can be taken
independently from its dynamical origin. It has an important link with
strata of Abelian differential (see :mod:`~sage.combinat.iet.strata`)
INPUT:
- ``intervals`` - string, two strings, list, tuples that can be converted to
two lists
- ``reduced`` - boolean (default: False) specifies reduction. False means
labelled permutation and True means reduced permutation.
- ``flips`` - iterable (default: None) the letters which correspond to
flipped intervals.
OUTPUT:
permutation -- the output type depends of the data.
EXAMPLES:
Creation of labelled permutations ::
sage: iet.Permutation('a b c d','d c b a')
a b c d
d c b a
sage: iet.Permutation([[0,1,2,3],[2,1,3,0]])
0 1 2 3
2 1 3 0
sage: iet.Permutation([0, 'A', 'B', 1], ['B', 0, 1, 'A'])
0 A B 1
B 0 1 A
Creation of reduced permutations::
sage: iet.Permutation('a b c', 'c b a', reduced = True)
a b c
c b a
sage: iet.Permutation([0, 1, 2, 3], [1, 3, 0, 2])
0 1 2 3
1 3 0 2
Creation of flipped permutations::
sage: iet.Permutation('a b c', 'c b a', flips=['a','b'])
-a -b c
c -b -a
sage: iet.Permutation('a b c', 'c b a', flips=['a'], reduced=True)
-a b c
c b -a
TESTS:
::
sage: p = iet.Permutation('a b c','c b a')
sage: iet.Permutation(p) == p
True
sage: iet.Permutation(p, reduced=True) == p.reduced()
True
::
sage: p = iet.Permutation('a','a',flips='a',reduced=True)
sage: iet.Permutation(p) == p
True
::
sage: p = iet.Permutation('a b c','c b a',flips='a')
sage: iet.Permutation(p) == p
True
sage: iet.Permutation(p, reduced=True) == p.reduced()
True
::
sage: p = iet.Permutation('a b c','c b a',reduced=True)
sage: iet.Permutation(p) == p
True
"""
from labelled import LabelledPermutation
from labelled import LabelledPermutationIET
from labelled import FlippedLabelledPermutationIET
from reduced import ReducedPermutation
from reduced import ReducedPermutationIET
from reduced import FlippedReducedPermutationIET
if 'reduced' not in kargs :
reduction = None
elif not isinstance(kargs["reduced"], bool) :
raise TypeError("reduced must be of type boolean")
else :
if kargs["reduced"] == True : reduction = True
else : reduction = False
if 'flips' not in kargs :
flips = []
else :
flips = list(kargs['flips'])
if 'alphabet' not in kargs :
alphabet = None
else :
alphabet = kargs['alphabet']
if len(args) == 1:
args = args[0]
if isinstance(args, LabelledPermutation):
if flips == []:
if reduction is None or reduction is False:
from copy import copy
return copy(args)
else:
return args.reduced()
else:
reduced = reduction in (None, False)
return PermutationIET(
args.list(),
reduced=reduced,
flips=flips,
alphabet=alphabet)
if isinstance(args, ReducedPermutation):
if flips == []:
if reduction is None or reduction is True:
from copy import copy
return copy(args)
else:
return PermutationIET(
args.list(),
reduced=True)
else:
reduced = reduction in (None, True)
return PermutationIET(
args.list(),
reduced=reduced,
flips=flips,
alphabet=alphabet)
a = _two_lists(args)
l = a[0] + a[1]
letters = set(l)
for letter in flips :
if letter not in letters :
raise ValueError, "flips contains not valid letters"
for letter in letters :
if a[0].count(letter) != 1 or a[1].count(letter) != 1:
raise ValueError, "letters must appear once in each interval"
if reduction == True :
if flips == [] :
return ReducedPermutationIET(a, alphabet=alphabet)
else :
return FlippedReducedPermutationIET(a, alphabet=alphabet, flips=flips)
else :
if flips == [] :
return LabelledPermutationIET(a, alphabet=alphabet)
else :
return FlippedLabelledPermutationIET(a, alphabet=alphabet, flips=flips)
def GeneralizedPermutation(*args,**kargs):
r"""
Returns a permutation of an interval exchange transformation.
Those permutations are the combinatoric part of linear involutions and were
introduced by Danthony-Nogueira [DN90]_. The full combinatoric study and
precise links with strata of quadratic differentials was achieved few years
later by Boissy-Lanneau [BL08]_.
INPUT:
- ``intervals`` - strings, list, tuples
- ``reduced`` - boolean (defaut: False) specifies reduction. False means
labelled permutation and True means reduced permutation.
- ``flips`` - iterable (default: None) the letters which correspond to
flipped intervals.
OUTPUT:
generalized permutation -- the output type depends on the data.
EXAMPLES:
Creation of labelled generalized permutations::
sage: iet.GeneralizedPermutation('a b b','c c a')
a b b
c c a
sage: iet.GeneralizedPermutation('a a','b b c c')
a a
b b c c
sage: iet.GeneralizedPermutation([[0,1,2,3,1],[4,2,5,3,5,4,0]])
0 1 2 3 1
4 2 5 3 5 4 0
Creation of reduced generalized permutations::
sage: iet.GeneralizedPermutation('a b b', 'c c a', reduced = True)
a b b
c c a
sage: iet.GeneralizedPermutation('a a b b', 'c c d d', reduced = True)
a a b b
c c d d
Creation of flipped generalized permutations::
sage: iet.GeneralizedPermutation('a b c a', 'd c d b', flips = ['a','b'])
-a -b c -a
d c d -b
"""
from template import FlippedPermutation
from labelled import LabelledPermutation
from labelled import LabelledPermutationLI
from labelled import FlippedLabelledPermutationLI
from reduced import ReducedPermutation
from reduced import ReducedPermutationLI
from reduced import FlippedReducedPermutationLI
if 'reduced' not in kargs :
reduction = None
elif not isinstance(kargs["reduced"], bool) :
raise TypeError("reduced must be of type boolean")
else :
if kargs["reduced"] == True : reduction = True
else : reduction = False
if 'flips' not in kargs :
flips = []
else :
flips = list(kargs['flips'])
if 'alphabet' not in kargs :
alphabet = None
else :
alphabet = kargs['alphabet']
if len(args) == 1:
args = args[0]
if isinstance(args, LabelledPermutation):
if flips == []:
if reduction is None or reduction is False:
from copy import copy
return copy(args)
else:
return args.reduced()
else:
reduced = reduction in (None, False)
return PermutationLI(
args.list(),
reduced=reduced,
flips=flips,
alphabet=alphabet)
if isinstance(args, ReducedPermutation):
if flips == []:
if reduction is None or reduction is True:
from copy import copy
return copy(args)
else:
return PermutationLI(
args.list(),
reduced=True)
else:
reduced = reduction in (None, True)
return PermutationLI(
args.list(),
reduced=reduced,
flips=flips,
alphabet=alphabet)
a = _two_lists(args)
if 'reduced' not in kargs :
reduction = False
elif not isinstance(kargs["reduced"], bool) :
raise TypeError("reduced must be of type boolean")
else :
if kargs["reduced"] == True : reduction = True
else : reduction = False
if 'flips' not in kargs :
flips = []
else :
flips = list(kargs['flips'])
if 'alphabet' not in kargs :
alphabet = None
else :
alphabet = kargs['alphabet']
l = a[0] + a[1]
letters = set(l)
for letter in flips :
if letter not in letters :
raise TypeError("The flip list is not valid")
for letter in letters :
if l.count(letter) != 2:
raise ValueError, "Letters must reappear twice"
b0 = a[0][:]
b1 = a[1][:]
for letter in letters :
if b0.count(letter) == 1 :
del b0[b0.index(letter)]
del b1[b1.index(letter)]
if (b0 == []) and (b1 == []):
return Permutation(a,**kargs)
elif (b0 == []) or (b1 == []):
raise ValueError, "There is no admissible length"
if reduction == True :
if flips == [] :
return ReducedPermutationLI(a, alphabet=alphabet)
else :
return FlippedReducedPermutationLI(a, alphabet=alphabet, flips=flips)
else :
if flips == [] :
return LabelledPermutationLI(a, alphabet=alphabet)
else :
return FlippedLabelledPermutationLI(a, alphabet=alphabet, flips=flips)
def Permutations_iterator(
nintervals=None,
irreducible=True,
reduced=False,
alphabet=None):
r"""
Returns an iterator over permutations.
This iterator allows you to iterate over permutations with given
constraints. If you want to iterate over permutations coming from a given
stratum you have to use the module :mod:`~sage.combinat.iet.strata` and
generate Rauzy diagrams from connected components.
INPUT:
- ``nintervals`` - non negative integer
- ``irreducible`` - boolean (default: True)
- ``reduced`` - boolean (default: False)
- ``alphabet`` - alphabet (default: None)
OUTPUT:
iterator -- an iterator over permutations
EXAMPLES:
Generates all reduced permutations with given number of intervals::
sage: P = iet.Permutations_iterator(nintervals=2,alphabet="ab",reduced=True)
sage: for p in P: print p, "\n* *"
a b
b a
* *
sage: P = iet.Permutations_iterator(nintervals=3,alphabet="abc",reduced=True)
sage: for p in P: print p, "\n* * *"
a b c
b c a
* * *
a b c
c a b
* * *
a b c
c b a
* * *
"""
from labelled import LabelledPermutationsIET_iterator
from reduced import ReducedPermutationsIET_iterator
if nintervals is None:
if alphabet is None:
raise ValueError, "You must specify an alphabet or a length"
else:
alphabet = Alphabet(alphabet)
if alphabet.cardinality() is Infinity:
raise ValueError, "You must sepcify a length with infinite alphabet"
nintervals = alphabet.cardinality()
elif alphabet is None:
alphabet = range(1,nintervals+1)
if reduced:
return ReducedPermutationsIET_iterator(
nintervals,
irreducible=irreducible,
alphabet=alphabet)
else:
return LabelledPermutationsIET_iterator(
nintervals,
irreducible=irreducible,
alphabet=alphabet)
def RauzyDiagram(*args, **kargs):
r"""
Return an object coding a Rauzy diagram.
The Rauzy diagram is an oriented graph with labelled edges. The set of
vertices corresponds to the permutations obtained by different operations
(mainly the .rauzy_move() operations that corresponds to an induction of
interval exchange transformation). The edges correspond to the action of the
different operations considered.
It first appeard in the original article of Rauzy [R79]_.
INPUT:
- ``intervals`` - lists, or strings, or tuples
- ``reduced`` - boolean (default: False) to precise reduction
- ``flips`` - list (default: []) for flipped permutations
- ``right_induction`` - boolean (default: True) consideration of left
induction in the diagram
- ``left_induction`` - boolean (default: False) consideration of right
induction in the diagram
- ``left_right_inversion`` - boolean (default: False) consideration of
inversion
- ``top_bottom_inversion`` - boolean (default: False) consideration of
reversion
- ``symmetric`` - boolean (default: False) consideration of the symmetric
operation
OUTPUT:
Rauzy diagram -- the Rauzy diagram that corresponds to your request
EXAMPLES:
Standard Rauzy diagrams::
sage: iet.RauzyDiagram('a b c d', 'd b c a')
Rauzy diagram with 12 permutations
sage: iet.RauzyDiagram('a b c d', 'd b c a', reduced = True)
Rauzy diagram with 6 permutations
Extended Rauzy diagrams::
sage: iet.RauzyDiagram('a b c d', 'd b c a', symmetric=True)
Rauzy diagram with 144 permutations
Using Rauzy diagrams and path in Rauzy diagrams::
sage: r = iet.RauzyDiagram('a b c', 'c b a')
sage: print r
Rauzy diagram with 3 permutations
sage: p = iet.Permutation('a b c','c b a')
sage: p in r
True
sage: g0 = r.path(p, 'top', 'bottom','top')
sage: g1 = r.path(p, 'bottom', 'top', 'bottom')
sage: print g0.is_loop(), g1.is_loop()
True True
sage: print g0.is_full(), g1.is_full()
False False
sage: g = g0 + g1
sage: g
Path of length 6 in a Rauzy diagram
sage: print g.is_loop(), g.is_full()
True True
sage: m = g.matrix()
sage: print m
[1 1 1]
[2 4 1]
[2 3 2]
sage: s = g.orbit_substitution()
sage: print s
WordMorphism: a->acbbc, b->acbbcbbc, c->acbc
sage: s.incidence_matrix() == m
True
We can then create the corresponding interval exchange transformation and
comparing the orbit of `0` to the fixed point of the orbit substitution::
sage: v = m.eigenvectors_right()[-1][1][0]
sage: T = iet.IntervalExchangeTransformation(p, v).normalize()
sage: print T
Interval exchange transformation of [0, 1[ with permutation
a b c
c b a
sage: w1 = []
sage: x = 0
sage: for i in range(20):
... w1.append(T.in_which_interval(x))
... x = T(x)
sage: w1 = Word(w1)
sage: w1
word: acbbcacbcacbbcbbcacb
sage: w2 = s.fixed_point('a')
sage: w2[:20]
word: acbbcacbcacbbcbbcacb
sage: w2[:20] == w1
True
"""
if not kargs.has_key('reduced'):
kargs['reduced'] = False
if not kargs.has_key('flips'):
kargs['flips'] =[]
if not kargs.has_key('alphabet'):
kargs['alphabet'] = None
p = GeneralizedPermutation(
args,
reduced= kargs['reduced'],
flips = kargs['flips'],
alphabet = kargs['alphabet'])
if not kargs.has_key('right_induction'):
kargs['right_induction'] = True
if not kargs.has_key('left_induction'):
kargs['left_induction'] = False
if not kargs.has_key('left_right_inversion'):
kargs['left_right_inversion'] = False
if not kargs.has_key('top_bottom_inversion'):
kargs['top_bottom_inversion'] = False
if not kargs.has_key('symmetric'):
kargs['symmetric'] = False
return p.rauzy_diagram(
right_induction = kargs['right_induction'],
left_induction = kargs['left_induction'],
left_right_inversion = kargs['left_right_inversion'],
top_bottom_inversion = kargs['top_bottom_inversion'],
symmetric = kargs['symmetric'])
def IntervalExchangeTransformation(permutation=None,lengths=None):
"""
Constructs an Interval exchange transformation.
An interval exchange transformation (or iet) is a map from an
interval to itself. It is defined on the interval except at a finite
number of points (the singularities) and is a translation on each
connected component of the complement of the singularities. Moreover it is a
bijection on its image (or it is injective).
An interval exchange transformation is encoded by two datas. A permutation
(that corresponds to the way we echange the intervals) and a vector of
positive reals (that corresponds to the lengths of the complement of the
singularities).
INPUT:
- ``permutation`` - a permutation
- ``lengths`` - a list or a dictionnary of lengths
OUTPUT:
interval exchange transformation -- an map of an interval
EXAMPLES:
Two initialization methods, the first using a iet.Permutation::
sage: p = iet.Permutation('a b c','c b a')
sage: t = iet.IntervalExchangeTransformation(p, {'a':1,'b':0.4523,'c':2.8})
The second is more direct::
sage: t = iet.IntervalExchangeTransformation(('a b','b a'),{'a':1,'b':4})
It's also possible to initialize the lengths only with a list::
sage: t = iet.IntervalExchangeTransformation(('a b c','c b a'),[0.123,0.4,2])
The two fundamental operations are Rauzy move and normalization::
sage: t = iet.IntervalExchangeTransformation(('a b c','c b a'),[0.123,0.4,2])
sage: s = t.rauzy_move()
sage: s_n = s.normalize(t.length())
sage: s_n.length() == t.length()
True
A not too simple example of a self similar interval exchange transformation::
sage: p = iet.Permutation('a b c d','d c b a')
sage: d = p.rauzy_diagram()
sage: g = d.path(p, 't', 't', 'b', 't', 'b', 'b', 't', 'b')
sage: m = g.matrix()
sage: v = m.eigenvectors_right()[-1][1][0]
sage: t = iet.IntervalExchangeTransformation(p,v)
sage: s = t.rauzy_move(iterations=8)
sage: s.normalize() == t.normalize()
True
"""
from iet import IntervalExchangeTransformation as _IET
from labelled import LabelledPermutationIET
from template import FlippedPermutation
if isinstance(permutation, FlippedPermutation):
raise TypeError, "flips are not yet implemented"
if isinstance(permutation, LabelledPermutationIET):
p = permutation
else:
p = Permutation(permutation,reduced=False)
if isinstance(lengths, dict):
l = [0] * len(p)
alphabet = p._alphabet
for letter in lengths:
l[alphabet.rank(letter)] = lengths[letter]
else:
l = list(lengths)
if len(l) != len(p):
raise ValueError, "bad number of lengths"
for x in l:
try:
y = float(x)
except ValueError:
raise TypeError, "unable to convert x (='%s') into a real number" %(str(x))
if y <= 0:
raise ValueError, "lengths must be positive"
return _IET(p,l)
IET = IntervalExchangeTransformation
