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):
- for connected stratum
representative()
- for hyperellitic component
representative()
- for non hyperelliptic component, the algorithm is the same as for connected component
- for odd component
representative()
- for even component
representative()
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 prescribed 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: a
H(1, 1)
sage: a.genus()
2
sage: a.nintervals()
5
sage: a = AbelianStratum(4,3,2,1)
sage: a
H(4, 3, 2, 1)
sage: a.genus()
6
sage: 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: a.connected_components()
[H_hyp(0)]
sage: a = AbelianStratum(6)
sage: cc = a.connected_components()
sage: cc
[H_hyp(6), H_odd(6), H_even(6)]
sage: for c in cc:
....: print(c)
....: print(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: 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: a == b
False
sage: a, a.connected_components()
(H^out(4, 2), [H_odd^out(4, 2), H_even^out(4, 2)])
sage: 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: 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: 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: 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: 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 = [x.rauzy_diagram().cardinality() for x in l]
sage: for c,i in zip(l,n):
....: print("{} : {}".format(c, i))
H_hyp^out(2) : 7
H_hyp^out(0, 0, 0) : 6
sage: sum(n)
13
sage: a = AbelianStrata(nintervals=5)
sage: l = sum([stratum.connected_components() for stratum in a], [])
sage: n = [x.rauzy_diagram().cardinality() for x in l]
sage: for c,i in zip(l,n):
....: print("{} : {}".format(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: sum(n)
71
sage: a = AbelianStrata(nintervals=6)
sage: l = sum([stratum.connected_components() for stratum in a], [])
sage: n = [x.rauzy_diagram().cardinality() for x in l]
sage: for c,i in zip(l,n):
....: print("{} : {}".format(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: sum(n)
461
-
sage.dynamics.flat_surfaces.strata.
AbelianStrata
(genus=None, nintervals=None, marked_separatrix=None)¶ Abelian strata.
INPUT:
genus
- a non negative integer orNone
nintervals
- a non negative integer orNone
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)¶ Bases:
sage.combinat.combinat.InfiniteAbstractCombinatorialClass
Abelian strata.
-
class
sage.dynamics.flat_surfaces.strata.
AbelianStrata_d
(nintervals=None, marked_separatrix=None)¶ Bases:
sage.combinat.combinat.CombinatorialClass
Strata with constraint number of intervals.
INPUT:
nintervals
- an integer greater than 1marked_separatrix
- ‘no’, ‘out’ or ‘in’
-
class
sage.dynamics.flat_surfaces.strata.
AbelianStrata_g
(genus=None, marked_separatrix=None)¶ Bases:
sage.combinat.combinat.CombinatorialClass
Stratas of genus g surfaces.
INPUT:
genus
- a non negative integermarked_separatrix
- ‘no’, ‘out’ or ‘in’
-
class
sage.dynamics.flat_surfaces.strata.
AbelianStrata_gd
(genus=None, nintervals=None, marked_separatrix=None)¶ Bases:
sage.combinat.combinat.CombinatorialClass
Abelian strata of prescribed genus and number of intervals.
INPUT:
genus
- integer: the genus of the surfacesnintervals
- integer: the number of intervalsmarked_separatrix
- ‘no’, ‘in’ or ‘out’
-
class
sage.dynamics.flat_surfaces.strata.
AbelianStratum
(*l, **d)¶ Bases:
sage.structure.sage_object.SageObject
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: 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: a H^in(4, 2) sage: b = AbelianStratum(2,4,marked_separatrix='in') sage: b H^in(2, 4) sage: a == b False
sage: a = AbelianStratum(4,2,marked_separatrix='out') sage: a H^out(4, 2) sage: b = AbelianStratum(2,4,marked_separatrix='out') sage: 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: a_hyp.representative() 1 2 3 4 5 6 7 7 6 5 4 3 2 1 sage: a_odd.representative() 0 1 2 3 4 5 6 3 2 4 6 5 1 0
You can choose the alphabet:
sage: 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 elseFalse
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
¶ alias of
ConnectedComponentOfAbelianStratum
-
class
sage.dynamics.flat_surfaces.strata.
ConnectedComponentOfAbelianStratum
(parent)¶ Bases:
sage.structure.sage_object.SageObject
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 permutationalphabet
- an alphabet orNone
: 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: c H_c(1, 1, 1, 1) sage: p = c.representative(alphabet=range(9)) sage: 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)¶ Bases:
sage.dynamics.flat_surfaces.strata.ConnectedComponentOfAbelianStratum
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)¶ Bases:
sage.dynamics.flat_surfaces.strata.ConnectedComponentOfAbelianStratum
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 (default:True
): whether you obtain a reduced or labelled permutationalphabet
- alphabet orNone
(default: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)¶ Bases:
sage.dynamics.flat_surfaces.strata.ConnectedComponentOfAbelianStratum
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)¶ Bases:
sage.dynamics.flat_surfaces.strata.ConnectedComponentOfAbelianStratum
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: 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: 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
-