r"""
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 :func:`Permutation`) some
associated graph (constructed using ?) and some maps
of intervals (constructed using :func:`IntervalExchangeTransformation`).
EXAMPLES::
sage: from surface_dynamics import *
Creation of an interval exchange transformation (iet)::
sage: T = iet.IntervalExchangeTransformation(('a b','b a'),(sqrt(2),1))
sage: 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: T
Interval exchange transformation of [0, 2[ with permutation
1 2 3
3 2 1
As the iet's are functions, you can compose and invert them::
sage: T = iet.IntervalExchangeTransformation(('a b','b a'),(sqrt(2),1))
sage: T*T
Interval exchange transformation of [0, sqrt(2) + 1[ with permutation
aa ab ba
ab ba aa
sage: S = T.inverse()
sage: S
Interval exchange transformation of [0, sqrt(2) + 1[ with permutation
b a
a b
sage: S * T
Interval exchange transformation of [0, sqrt(2) + 1[ with permutation
aa bb
aa bb
sage: (S * T).is_identity()
True
sage: T * S
Interval exchange transformation of [0, sqrt(2) + 1[ with permutation
bb aa
bb aa
sage: (T * S).is_identity()
True
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: 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: 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: p
a a b
b c c
By default, the permutations are labelled (it means that the labels are
important and (a b / b a) differs from (b a / a b)). It sometimes useful to deal
with reduced permutations for which the order does not import::
sage: p = iet.Permutation('a b c', 'c b a', reduced = True)
sage: p
a b c
c b a
Permutations with flips::
sage: p1 = iet.Permutation('a b c', 'c b a', flips = ['a','c'])
sage: 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 default, 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
AUTHORS:
- Vincent Delecroix (2009-09-29): initial version
"""
# ****************************************************************************
# Copyright (C) 2019-2021 Vincent Delecroix <20100.delecroix@gmail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
# as published by the Free Software Foundation; either version 2 of
# the License, or (at your option) any later version.
# https://www.gnu.org/licenses/
# ****************************************************************************
from __future__ import print_function, absolute_import
from six.moves import range
from sage.combinat.words.alphabet import Alphabet
from sage.rings.infinity import Infinity
from sage.misc.decorators import rename_keyword
def _two_lists(arg1, arg2):
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 surface_dynamics.interval_exchanges.constructors import _two_lists
sage: _two_lists(('a1 a2','b1 b2'), None)
[['a1', 'a2'], ['b1', 'b2']]
sage: _two_lists('a1 a2\nb1 b2', None)
[['a1', 'a2'], ['b1', 'b2']]
sage: _two_lists(['a b','c'], None)
[['a', 'b'], ['c']]
Make sure that tuples are properly converted::
sage: _two_lists(((0, 1), (1, 0)), None)
[[0, 1], [1, 0]]
sage: _two_lists((0, 1), (1, 0))
[[0, 1], [1, 0]]
ValueError or TypeError are raised with invalid arguments::
sage: _two_lists('a b', None)
Traceback (most recent call last):
...
ValueError: the chain must contain two lines
sage: _two_lists('a b\nc d\ne f', None)
Traceback (most recent call last):
...
ValueError: the chain must contain two lines
sage: _two_lists(1, 1)
Traceback (most recent call last):
...
TypeError: argument not accepted
"""
from sage.combinat.permutation import Permutation as CombinatPermutation
from sage.groups.perm_gps.permgroup_element import PermutationGroupElement
if arg2 is None:
if isinstance(arg1, str):
t = arg1.split('\n')
if len(t) != 2:
raise ValueError("the chain must contain two lines")
return [t[0].split(), t[1].split()]
elif isinstance(arg1, CombinatPermutation):
return [list(range(1, len(arg1) + 1)), list(arg1)]
elif isinstance(arg1, PermutationGroupElement):
dom = list(arg1.parent().domain())
return [dom, [arg1(i) for i in dom]]
elif isinstance(arg1, (tuple, list)):
try:
t = CombinatPermutation(arg1)
except Exception:
if len(arg1) != 2:
raise ValueError('argument not accepted')
arg1, arg2 = arg1
else:
return [list(range(1, len(t) + 1)), list(t)]
if arg2 is None:
raise ValueError("argument can not be split into two parts")
res = []
for a in (arg1, arg2):
if isinstance(a, (tuple, list)):
res.append(list(a))
elif isinstance(a, str):
res.append(a.split())
else:
raise TypeError('argument not accepted')
return res
[docs]
def Permutation(arg1, arg2=None, reduced=None, flips=None, alphabet=None):
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 [Rau80]_ and William Veech [Vee78]_.
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 ?)
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.
- ``alphabet`` - (optional)
OUTPUT:
permutation -- the output type depends of the data.
EXAMPLES::
sage: from surface_dynamics import *
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], reduced=True)
0 1 2 3
1 3 0 2
sage: iet.Permutation([2,1], reduced=True)
1 2
2 1
Managing the alphabet: two labelled permutations with different (ordered)
alphabet but with the same labels are different::
sage: p = iet.Permutation('a b','b a', alphabet='ab')
sage: q = iet.Permutation('a b','b a', alphabet='ba')
sage: str(p) == str(q)
True
sage: p == q
False
sage: p.rauzy_move_matrix('top')
[1 0]
[1 1]
sage: q.rauzy_move_matrix('top')
[1 1]
[0 1]
For reduced permutations, the alphabet does not play any role excepted for
printing the object::
sage: p = iet.Permutation('a b c','c b a', reduced=True)
sage: q = iet.Permutation([0,1,2],[2,1,0], reduced=True)
sage: p == q
True
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='ab', reduced=True)
-a -b c
c -b -a
TESTS::
sage: type(iet.Permutation('a b c', 'c b a', reduced=True))
<class 'surface_dynamics.interval_exchanges.reduced.ReducedPermutationIET'>
sage: type(iet.Permutation('a b c', 'c b a', reduced=False))
<class 'surface_dynamics.interval_exchanges.labelled.LabelledPermutationIET'>
sage: type(iet.Permutation('a b c', 'c b a', reduced=True, flips=['a','b']))
<class 'surface_dynamics.interval_exchanges.reduced.FlippedReducedPermutationIET'>
sage: type(iet.Permutation('a b c', 'c b a', reduced=False, flips=['a','b']))
<class 'surface_dynamics.interval_exchanges.labelled.FlippedLabelledPermutationIET'>
sage: p = iet.Permutation(('a b c','c b a'))
sage: iet.Permutation(p) == p
True
sage: q = iet.Permutation(p, reduced=True)
sage: q == p
False
sage: q == 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
"""
if arg2 is None:
from .template import Permutation as Permutation_class
if isinstance(arg1, Permutation_class):
return Permutation(
arg1.list(),
reduced=(arg1._labels is None) if reduced is None else reduced,
flips=arg1.flips() if flips is None else flips,
alphabet=arg1.alphabet() if alphabet is None else alphabet)
if reduced is None:
reduced = False
a = _two_lists(arg1, arg2)
l = a[0] + a[1]
letters = set(l)
if flips is not None:
# make it so that no flip is equivalent to not specifying the flips
if not flips:
flips = None
else:
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 reduced:
if flips is None:
from .reduced import ReducedPermutationIET as cls
else:
from .reduced import FlippedReducedPermutationIET as cls
else:
if flips is None:
from .labelled import LabelledPermutationIET as cls
else:
from .labelled import FlippedLabelledPermutationIET as cls
return cls(a, alphabet=alphabet, reduced=reduced, flips=flips)
[docs]
def GeneralizedPermutation(arg1, arg2=None, reduced=None, flips=None, alphabet=None):
r"""
Returns a permutation of an interval exchange transformation.
Those permutations are the combinatoric part of linear involutions and were
introduced by Danthony-Nogueira [DanNog90]_ (the flipped version is also
considered in [Nog89]_). The full combinatoric study and precise links with
strata of quadratic differentials was achieved few years later by
Boissy-Lanneau [BoiLan09]_.
INPUT:
- ``intervals`` - strings, list, tuples
- ``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:
generalized permutation -- the output type depends on the data.
EXAMPLES::
sage: from surface_dynamics import *
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
TESTS::
sage: type(iet.GeneralizedPermutation('a b b', 'c c a', reduced=True))
<class 'surface_dynamics.interval_exchanges.reduced.ReducedPermutationLI'>
sage: type(iet.GeneralizedPermutation('a b b', 'c c a', reduced=False))
<class 'surface_dynamics.interval_exchanges.labelled.LabelledPermutationLI'>
sage: type(iet.GeneralizedPermutation('a b b', 'c c a', reduced=True, flips=['a','b']))
<class 'surface_dynamics.interval_exchanges.reduced.FlippedReducedPermutationLI'>
sage: type(iet.GeneralizedPermutation('a b b', 'c c a', reduced=False, flips=['a','b']))
<class 'surface_dynamics.interval_exchanges.labelled.FlippedLabelledPermutationLI'>
"""
if arg2 is None:
from .template import Permutation as Permutation_class
if isinstance(arg1, Permutation_class):
return GeneralizedPermutation(
arg1.list(),
reduced=(arg1._labels is None) if reduced is None else reduced,
flips=arg1.flips() if flips is None else flips,
alphabet=arg1.alphabet() if alphabet is None else alphabet)
if reduced is None:
reduced = False
a = _two_lists(arg1, arg2)
l = a[0] + a[1]
letters = set(l)
if flips is not None:
for letter in flips:
if letter not in letters:
raise ValueError("flips contains not valid letters")
for letter in letters:
if l.count(letter) != 2:
raise ValueError("letters must appear 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 not b0 and not b1:
return Permutation(a[0], a[1], reduced=reduced, flips=flips, alphabet=alphabet)
elif not b0 or not b1:
raise ValueError("no admissible length")
if reduced:
if flips is None:
from .reduced import ReducedPermutationLI as cls
else:
from .reduced import FlippedReducedPermutationLI as cls
else:
if flips is None:
from .labelled import LabelledPermutationLI as cls
else:
from .labelled import FlippedLabelledPermutationLI as cls
return cls(a, alphabet=alphabet, reduced=reduced, flips=flips)
[docs]
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 ? 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::
sage: from surface_dynamics import *
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("%s\n* *" % p)
a b
b a
* *
sage: P = iet.Permutations_iterator(nintervals=3,alphabet="abc",reduced=True)
sage: for p in P: print("%s\n* * *" % p)
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 specify 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)
[docs]
@rename_keyword(
lr_inversion='left_right_inversion',
tb_inversion='top_bottom_inversion')
def RauzyDiagram(arg1, arg2=None, reduced=False, flips=None, alphabet=None,
right_induction=True, left_induction=False,
left_right_inversion=False,
top_bottom_inversion=False,
symmetric=False):
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 [Rau80]_.
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::
sage: from surface_dynamics import *
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: 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: g0.is_loop()
True
sage: g1.is_loop()
True
sage: g0.is_full()
False
sage: g1.is_full()
False
sage: g = g0 + g1
sage: g
Path of length 6 in a Rauzy diagram
sage: g.is_loop()
True
sage: g.is_full()
True
sage: m = g.matrix()
sage: m
[1 1 1]
[2 4 1]
[2 3 2]
sage: s = g.orbit_substitution()
sage: print(s)
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
"""
p = GeneralizedPermutation(
arg1, arg2,
reduced=reduced,
flips=flips,
alphabet=alphabet)
return p.rauzy_diagram(
right_induction=right_induction,
left_induction=left_induction,
left_right_inversion=left_right_inversion,
top_bottom_inversion=top_bottom_inversion,
symmetric=symmetric)
# TODO
# def GeneralizedPermutation_iterator():
# pass
IET = IntervalExchangeTransformation
IETFamily = IntervalExchangeTransformationFamily
[docs]
def FlipSequence(*args, **kwds):
r"""
Build a flip sequence corresponding to a path in a Rauzy diagram.
A flip sequence is built from an initial permutation (built from :func:`Permutation` or
:func:`GeneralizedPermutation`) and a sequence of Rauzy inductions that could be of the
following form:
- a letter ``'t'`` or ``'b'`` for top right and bottom right induction
respectively
- a pair of letters ``'tr'``, ``'br'``, ``'tl'`` or ``'bl'`` for top right,
bottom right, top left and bottom left induction respectively
For all available options, see :class:`~surface_dynamics.interval_exchanges.flip_sequence.IETFlipSequence`.
EXAMPLES::
sage: from surface_dynamics import iet
sage: p = iet.Permutation('a b', 'b a')
sage: f = iet.FlipSequence(p, ['t', 'b'])
sage: f.substitution()
WordMorphism: a->ab, b->abb
sage: f = iet.FlipSequence(p, ['tr', 'bl'])
sage: f.substitution()
WordMorphism: a->bab, b->b
"""
from .flip_sequence import IETFlipSequence
return IETFlipSequence(*args, **kwds)