# 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 iet.Permutation()) some associated graph (constructed using iet.RauzyGraph()) and some maps of intervals (constructed using 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 theoretic 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] (1, 2) 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
sage.dynamics.interval_exchanges.constructors.GeneralizedPermutation(*args, **kargs)

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

TESTS:

sage: iet.GeneralizedPermutation('a a b b', 'c c d d', reduced = 'may')
Traceback (most recent call last):
...
TypeError: reduced must be of type boolean
sage: iet.GeneralizedPermutation('a b c a', 'd c d b', flips = ['e','b'])
Traceback (most recent call last):
...
TypeError: The flip list is not valid
sage: iet.GeneralizedPermutation('a b c a', 'd c c b', flips = ['a','b'])
Traceback (most recent call last):
...
ValueError: Letters must reappear twice
sage.dynamics.interval_exchanges.constructors.IET(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

TESTS:

sage: iet.IntervalExchangeTransformation(('a b c','c b a'),[0.123,2])
Traceback (most recent call last):
...
sage: iet.IntervalExchangeTransformation(('a b c','c b a'),[0.1,'rho',2])
Traceback (most recent call last):
...
TypeError: unable to convert x (='rho') into a real number
sage: iet.IntervalExchangeTransformation(('a b c','c b a'),[0.1,-2,2])
Traceback (most recent call last):
...
ValueError: lengths must be positive
sage.dynamics.interval_exchanges.constructors.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

TESTS:

sage: iet.IntervalExchangeTransformation(('a b c','c b a'),[0.123,2])
Traceback (most recent call last):
...
sage: iet.IntervalExchangeTransformation(('a b c','c b a'),[0.1,'rho',2])
Traceback (most recent call last):
...
TypeError: unable to convert x (='rho') into a real number
sage: iet.IntervalExchangeTransformation(('a b c','c b a'),[0.1,-2,2])
Traceback (most recent call last):
...
ValueError: lengths must be positive
sage.dynamics.interval_exchanges.constructors.Permutation(*args, **kargs)

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 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

TESTS:

sage: iet.Permutation('a b c','c b a',reduced='badly')
Traceback (most recent call last):
...
TypeError: reduced must be of type boolean
sage: iet.Permutation('a','a',flips='b',reduced=True)
Traceback (most recent call last):
...
ValueError: flips contains not valid letters
sage: iet.Permutation('a b c','c a a',reduced=True)
Traceback (most recent call last):
...
ValueError: letters must appear once in each interval
sage.dynamics.interval_exchanges.constructors.Permutations_iterator(nintervals=None, irreducible=True, reduced=False, alphabet=None)

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 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
* * *

TESTS:

sage: P = iet.Permutations_iterator(nintervals=None, alphabet=None)
Traceback (most recent call last):
...
ValueError: You must specify an alphabet or a length
sage: P = iet.Permutations_iterator(nintervals=None, alphabet=ZZ)
Traceback (most recent call last):
...
ValueError: You must specify a length with infinite alphabet
sage.dynamics.interval_exchanges.constructors.RauzyDiagram(*args, **kargs)

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: 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

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Interval exchange transformations and linear involutions

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Labelled permutations