# Strata of differentials on Riemann surfaces¶

The space of Abelian (or quadratic) differentials is stratified by the degrees of the zeroes (and simple poles for quadratic differentials). Each stratum has one, two or three connected components and each is associated to an (extended) Rauzy class. The connected_components() method (only available for Abelian stratum) give the decomposition of a stratum (which corresponds to the SAGE object AbelianStratum).

The work for Abelian differentials was done by Maxim Kontsevich and Anton Zorich in [KonZor03] and for quadratic differentials by Erwan Lanneau in [Lan08]. Zorich gave an algorithm to pass from a connected component of a stratum to the associated Rauzy class (for both interval exchange transformations and linear involutions) in [Zor08] and is implemented for Abelian stratum at different level (approximately one for each component):

The inverse operation (pass from an interval exchange transformation to the connected component) is partially written in [KonZor03] and simply named here connected_component().

All the code here was first available on Mathematica [ZS].

REFERENCES:

 [KonZor03] (1, 2) M. Kontsevich, A. Zorich “Connected components of the moduli space of Abelian differentials with prescripebd singularities” Invent. math. 153, 631-678 (2003)
 [Lan08] E. Lanneau “Connected components of the strata of the moduli spaces of quadratic differentials”, Annales sci. de l’ENS, serie 4, fascicule 1, 41, 1-56 (2008)
 [Zor08] (1, 2, 3, 4, 5) A. Zorich “Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials”, Journal of Modern Dynamics, vol. 2, no 1, 139-185 (2008) (http://www.math.psu.edu/jmd)
 [ZS] Anton Zorich, “Generalized Permutation software” (http://perso.univ-rennes1.fr/anton.zorich/Software/software_en.html)

Note

The quadratic strata are not yet implemented.

AUTHORS:

• Vincent Delecroix (2009-09-29): initial version

EXAMPLES:

Construction of a stratum from a list of singularity degrees:

sage: a = AbelianStratum(1,1)
sage: print a
H(1, 1)
sage: print a.genus()
2
sage: print a.nintervals()
5

sage: a = AbelianStratum(4,3,2,1)
sage: print a
H(4, 3, 2, 1)
sage: print a.genus()
6
sage: print a.nintervals()
15


By convention, the degrees are always written in decreasing order:

sage: a1 = AbelianStratum(4,3,2,1)
sage: a1
H(4, 3, 2, 1)
sage: a2 = AbelianStratum(2,3,1,4)
sage: a2
H(4, 3, 2, 1)
sage: a1 == a2
True


It is also possible to consider stratum with an incoming or an outgoing separatrix marked (the aim of this consideration is to attach a specified degree at the left or the right of the associated interval exchange transformation):

sage: a_out = AbelianStratum(1, 1, marked_separatrix='out')
sage: a_out
H^out(1, 1)
sage: a_in = AbelianStratum(1, 1, marked_separatrix='in')
sage: a_in
H^in(1, 1)
sage: a_out == a_in
False


Get a list of strata with constraints on genus or on the number of intervals of a representative:

sage: for a in AbelianStrata(genus=3):
....:     print a
H(4)
H(3, 1)
H(2, 2)
H(2, 1, 1)
H(1, 1, 1, 1)

sage: for a in AbelianStrata(nintervals=5):
....:     print a
H^out(0, 2)
H^out(2, 0)
H^out(1, 1)
H^out(0, 0, 0, 0)

sage: for a in AbelianStrata(genus=2, nintervals=5):
....:     print a
H^out(0, 2)
H^out(2, 0)
H^out(1, 1)


Obtains the connected components of a stratum:

sage: a = AbelianStratum(0)
sage: print a.connected_components()
[H_hyp(0)]

sage: a = AbelianStratum(6)
sage: cc = a.connected_components()
sage: print cc
[H_hyp(6), H_odd(6), H_even(6)]
sage: for c in cc:
....:     print c, "\n", c.representative(alphabet=range(1,9))
H_hyp(6)
1 2 3 4 5 6 7 8
8 7 6 5 4 3 2 1
H_odd(6)
1 2 3 4 5 6 7 8
4 3 6 5 8 7 2 1
H_even(6)
1 2 3 4 5 6 7 8
6 5 4 3 8 7 2 1

sage: a = AbelianStratum(1, 1, 1, 1)
sage: print a.connected_components()
[H_c(1, 1, 1, 1)]
sage: c = a.connected_components()[0]
sage: print c.representative(alphabet="abcdefghi")
a b c d e f g h i
e d c f i h g b a


The zero attached on the left of the associated Abelian permutation corresponds to the first singularity degree:

sage: a = AbelianStratum(4, 2, marked_separatrix='out')
sage: b = AbelianStratum(2, 4, marked_separatrix='out')
sage: print a == b
False
sage: print a, ":", a.connected_components()
H^out(4, 2) : [H_odd^out(4, 2), H_even^out(4, 2)]
sage: print b, ":", b.connected_components()
H^out(2, 4) : [H_odd^out(2, 4), H_even^out(2, 4)]
sage: a_odd, a_even = a.connected_components()
sage: b_odd, b_even = b.connected_components()


The representatives are hence different:

sage: print a_odd.representative(alphabet=range(1,10))
1 2 3 4 5 6 7 8 9
4 3 6 5 7 9 8 2 1
sage: print b_odd.representative(alphabet=range(1,10))
1 2 3 4 5 6 7 8 9
4 3 5 7 6 9 8 2 1

sage: print a_even.representative(alphabet=range(1,10))
1 2 3 4 5 6 7 8 9
6 5 4 3 7 9 8 2 1
sage: print b_even.representative(alphabet=range(1,10))
1 2 3 4 5 6 7 8 9
7 6 5 4 3 9 8 2 1


You can retrieve the decomposition of the irreducible Abelian permutations into Rauzy diagrams from the classification of strata:

sage: a = AbelianStrata(nintervals=4)
sage: l = sum([stratum.connected_components() for stratum in a], [])
sage: n = map(lambda x: x.rauzy_diagram().cardinality(), l)
sage: for c,i in zip(l,n):
....:     print c, ":", i
H_hyp^out(2) : 7
H_hyp^out(0, 0, 0) : 6
sage: print sum(n)
13

sage: a = AbelianStrata(nintervals=5)
sage: l = sum([stratum.connected_components() for stratum in a], [])
sage: n = map(lambda x: x.rauzy_diagram().cardinality(), l)
sage: for c,i in zip(l,n):
....:     print c, ":", i
H_hyp^out(0, 2) : 11
H_hyp^out(2, 0) : 35
H_hyp^out(1, 1) : 15
H_hyp^out(0, 0, 0, 0) : 10
sage: print sum(n)
71

sage: a = AbelianStrata(nintervals=6)
sage: l = sum([stratum.connected_components() for stratum in a], [])
sage: n = map(lambda x: x.rauzy_diagram().cardinality(), l)
sage: for c,i in zip(l,n):
....:     print c, ":", i
H_hyp^out(4) : 31
H_odd^out(4) : 134
H_hyp^out(0, 2, 0) : 66
H_hyp^out(2, 0, 0) : 105
H_hyp^out(0, 1, 1) : 20
H_hyp^out(1, 1, 0) : 90
H_hyp^out(0, 0, 0, 0, 0) : 15
sage: print sum(n)
461

sage.dynamics.flat_surfaces.strata.AbelianStrata(genus=None, nintervals=None, marked_separatrix=None)

Abelian strata.

INPUT:

• genus - a non negative integer or None
• nintervals - a non negative integer or None
• marked_separatrix - ‘no’ (for no marking), ‘in’ (for marking an incoming separatrix) or ‘out’ (for marking an outgoing separatrix)

EXAMPLES:

Abelian strata with a given genus:

sage: for s in AbelianStrata(genus=1): print s
H(0)

sage: for s in AbelianStrata(genus=2): print s
H(2)
H(1, 1)

sage: for s in AbelianStrata(genus=3): print s
H(4)
H(3, 1)
H(2, 2)
H(2, 1, 1)
H(1, 1, 1, 1)

sage: for s in AbelianStrata(genus=4): print s
H(6)
H(5, 1)
H(4, 2)
H(4, 1, 1)
H(3, 3)
H(3, 2, 1)
H(3, 1, 1, 1)
H(2, 2, 2)
H(2, 2, 1, 1)
H(2, 1, 1, 1, 1)
H(1, 1, 1, 1, 1, 1)


Abelian strata with a given number of intervals:

sage: for s in AbelianStrata(nintervals=2): print s
H^out(0)

sage: for s in AbelianStrata(nintervals=3): print s
H^out(0, 0)

sage: for s in AbelianStrata(nintervals=4): print s
H^out(2)
H^out(0, 0, 0)

sage: for s in AbelianStrata(nintervals=5): print s
H^out(0, 2)
H^out(2, 0)
H^out(1, 1)
H^out(0, 0, 0, 0)


Abelian strata with both constraints:

sage: for s in AbelianStrata(genus=2, nintervals=4): print s
H^out(2)

sage: for s in AbelianStrata(genus=5, nintervals=12): print s
H^out(8, 0, 0)
H^out(0, 8, 0)
H^out(0, 7, 1)
H^out(1, 7, 0)
H^out(7, 1, 0)
H^out(0, 6, 2)
H^out(2, 6, 0)
H^out(6, 2, 0)
H^out(1, 6, 1)
H^out(6, 1, 1)
H^out(0, 5, 3)
H^out(3, 5, 0)
H^out(5, 3, 0)
H^out(1, 5, 2)
H^out(2, 5, 1)
H^out(5, 2, 1)
H^out(0, 4, 4)
H^out(4, 4, 0)
H^out(1, 4, 3)
H^out(3, 4, 1)
H^out(4, 3, 1)
H^out(2, 4, 2)
H^out(4, 2, 2)
H^out(2, 3, 3)
H^out(3, 3, 2)

class sage.dynamics.flat_surfaces.strata.AbelianStrata_all(category=None, *keys, **opts)

Abelian strata.

class sage.dynamics.flat_surfaces.strata.AbelianStrata_d(nintervals=None, marked_separatrix=None)

Strata with constraint number of intervals.

INPUT:

• nintervals - an integer greater than 1
• marked_separatrix - ‘no’, ‘out’ or ‘in’
class sage.dynamics.flat_surfaces.strata.AbelianStrata_g(genus=None, marked_separatrix=None)

Stratas of genus g surfaces.

INPUT:

• genus - a non negative integer
• marked_separatrix - ‘no’, ‘out’ or ‘in’
class sage.dynamics.flat_surfaces.strata.AbelianStrata_gd(genus=None, nintervals=None, marked_separatrix=None)

Abelian strata of prescribed genus and number of intervals.

INPUT:

• genus - integer: the genus of the surfaces
• nintervals - integer: the number of intervals
• marked_separatrix - ‘no’, ‘in’ or ‘out’
class sage.dynamics.flat_surfaces.strata.AbelianStratum(*l, **d)

Stratum of Abelian differentials.

A stratum with a marked outgoing separatrix corresponds to Rauzy diagram with left induction, a stratum with marked incoming separatrix correspond to Rauzy diagram with right induction. If there is no marked separatrix, the associated Rauzy diagram is the extended Rauzy diagram (consideration of the sage.dynamics.interval_exchanges.template.Permutation.symmetric() operation of Boissy-Lanneau).

When you want to specify a marked separatrix, the degree on which it is is the first term of your degrees list.

INPUT:

• marked_separatrix - None (default) or ‘in’ (for incoming separatrix) or ‘out’ (for outgoing separatrix).

EXAMPLES:

Creation of an Abelian stratum and get its connected components:

sage: a = AbelianStratum(2, 2)
sage: print a
H(2, 2)
sage: a.connected_components()
[H_hyp(2, 2), H_odd(2, 2)]


Specification of marked separatrix:

sage: a = AbelianStratum(4,2,marked_separatrix='in')
sage: print a
H^in(4, 2)
sage: b = AbelianStratum(2,4,marked_separatrix='in')
sage: print b
H^in(2, 4)
sage: a == b
False

sage: a = AbelianStratum(4,2,marked_separatrix='out')
sage: print a
H^out(4, 2)
sage: b = AbelianStratum(2,4,marked_separatrix='out')
sage: print b
H^out(2, 4)
sage: a == b
False


Get a representative of a connected component:

sage: a = AbelianStratum(2,2)
sage: a_hyp, a_odd = a.connected_components()
sage: print a_hyp.representative()
1 2 3 4 5 6 7
7 6 5 4 3 2 1
sage: print a_odd.representative()
0 1 2 3 4 5 6
3 2 4 6 5 1 0


You can choose the alphabet:

sage: print a_odd.representative(alphabet="ABCDEFGHIJKLMNOPQRSTUVWXYZ")
A B C D E F G
D C E G F B A


By default, you get a reduced permutation, but you can specify that you want a labelled one:

sage: p_reduced = a_odd.representative()
sage: p_labelled = a_odd.representative(reduced=False)

connected_components()

Lists the connected components of the Stratum.

OUTPUT:

list – a list of connected components of stratum

EXAMPLES:

sage: AbelianStratum(0).connected_components()
[H_hyp(0)]

sage: AbelianStratum(2).connected_components()
[H_hyp(2)]
sage: AbelianStratum(1,1).connected_components()
[H_hyp(1, 1)]

sage: AbelianStratum(4).connected_components()
[H_hyp(4), H_odd(4)]
sage: AbelianStratum(3,1).connected_components()
[H_c(3, 1)]
sage: AbelianStratum(2,2).connected_components()
[H_hyp(2, 2), H_odd(2, 2)]
sage: AbelianStratum(2,1,1).connected_components()
[H_c(2, 1, 1)]
sage: AbelianStratum(1,1,1,1).connected_components()
[H_c(1, 1, 1, 1)]

genus()

Returns the genus of the stratum.

OUTPUT:

integer – the genus

EXAMPLES:

sage: AbelianStratum(0).genus()
1
sage: AbelianStratum(1,1).genus()
2
sage: AbelianStratum(3,2,1).genus()
4

is_connected()

Tests if the strata is connected.

OUTPUT:

boolean – True if it is connected else False

EXAMPLES:

sage: AbelianStratum(2).is_connected()
True
sage: AbelianStratum(2).connected_components()
[H_hyp(2)]

sage: AbelianStratum(2,2).is_connected()
False
sage: AbelianStratum(2,2).connected_components()
[H_hyp(2, 2), H_odd(2, 2)]

nintervals()

Returns the number of intervals of any iet of the strata.

OUTPUT:

integer – the number of intervals for any associated iet

EXAMPLES:

sage: AbelianStratum(0).nintervals()
2
sage: AbelianStratum(0,0).nintervals()
3
sage: AbelianStratum(2).nintervals()
4
sage: AbelianStratum(1,1).nintervals()
5

sage.dynamics.flat_surfaces.strata.CCA
class sage.dynamics.flat_surfaces.strata.ConnectedComponentOfAbelianStratum(parent)

Connected component of Abelian stratum.

Warning

Internal class! Do not use directly!

TESTS:

Tests for outgoing marked separatrices:

sage: a = AbelianStratum(4,2,0,marked_separatrix='out')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_out_degree()
4
sage: a_even.representative().attached_out_degree()
4

sage: a = AbelianStratum(2,4,0,marked_separatrix='out')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_out_degree()
2
sage: a_even.representative().attached_out_degree()
2

sage: a = AbelianStratum(0,4,2,marked_separatrix='out')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_out_degree()
0
sage: a_even.representative().attached_out_degree()
0

sage: a = AbelianStratum(3,2,1,marked_separatrix='out')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_out_degree()
3

sage: a = AbelianStratum(2,3,1,marked_separatrix='out')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_out_degree()
2

sage: a = AbelianStratum(1,3,2,marked_separatrix='out')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_out_degree()
1


Tests for incoming separatrices:

sage: a = AbelianStratum(4,2,0,marked_separatrix='in')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_in_degree()
4
sage: a_even.representative().attached_in_degree()
4

sage: a = AbelianStratum(2,4,0,marked_separatrix='in')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_in_degree()
2
sage: a_even.representative().attached_in_degree()
2

sage: a = AbelianStratum(0,4,2,marked_separatrix='in')
sage: a_odd, a_even = a.connected_components()
sage: a_odd.representative().attached_in_degree()
0
sage: a_even.representative().attached_in_degree()
0

sage: a = AbelianStratum(3,2,1,marked_separatrix='in')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_in_degree()
3

sage: a = AbelianStratum(2,3,1,marked_separatrix='in')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_in_degree()
2

sage: a = AbelianStratum(1,3,2,marked_separatrix='in')
sage: a_c = a.connected_components()[0]
sage: a_c.representative().attached_in_degree()
1

genus()

Returns the genus of the surfaces in this connected component.

OUTPUT:

integer – the genus of the surface

EXAMPLES:

sage: a = AbelianStratum(6,4,2,0,0)
sage: c_odd, c_even = a.connected_components()
sage: c_odd.genus()
7
sage: c_even.genus()
7

sage: a = AbelianStratum([1]*8)
sage: c = a.connected_components()[0]
sage: c.genus()
5

nintervals()

Returns the number of intervals of the representative.

OUTPUT:

integer – the number of intervals in any representative

EXAMPLES:

sage: a = AbelianStratum(6,4,2,0,0)
sage: c_odd, c_even = a.connected_components()
sage: c_odd.nintervals()
18
sage: c_even.nintervals()
18

sage: a = AbelianStratum([1]*8)
sage: c = a.connected_components()[0]
sage: c.nintervals()
17

parent()

The stratum of this component

OUTPUT:

stratum - the stratum where this component leaves

EXAMPLES:

sage: p = iet.Permutation('a b','b a')
sage: c = p.connected_component()
sage: c.parent()
H(0)

rauzy_diagram(reduced=True)

Returns the Rauzy diagram associated to this connected component.

OUTPUT:

rauzy diagram – the Rauzy diagram associated to this stratum

EXAMPLES:

sage: c = AbelianStratum(0).connected_components()[0]
sage: r = c.rauzy_diagram()

representative(reduced=True, alphabet=None)

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)

sage.dynamics.flat_surfaces.strata.EvenCCA
class sage.dynamics.flat_surfaces.strata.EvenConnectedComponentOfAbelianStratum(parent)

Connected component of Abelian stratum with even spin structure.

Warning

Internal class! Do not use directly!

representative(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: 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)

sage.dynamics.flat_surfaces.strata.HypCCA
class sage.dynamics.flat_surfaces.strata.HypConnectedComponentOfAbelianStratum(parent)

Hyperelliptic component of Abelian stratum.

Warning

Internal class! Do not use directly!

representative(reduced=True, alphabet=None)

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)

sage.dynamics.flat_surfaces.strata.NonHypCCA
class sage.dynamics.flat_surfaces.strata.NonHypConnectedComponentOfAbelianStratum(parent)

Non hyperelliptic component of Abelian stratum.

Warning

Internal class! Do not use directly!

sage.dynamics.flat_surfaces.strata.OddCCA
class sage.dynamics.flat_surfaces.strata.OddConnectedComponentOfAbelianStratum(parent)

Connected component of an Abelian stratum with odd spin parity.

Warning

Internal class! Do not use directly!

representative(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


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