abelian_strata#
Strata of differentials on Riemann surfaces
The space of Abelian differentials on Riemann surfaces of a given genus is
stratified by degrees of zeros. Each stratum has one, two or three connected
components each of which is associated to an extended Rauzy class. The
components() method
lists connected components of a stratum.
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
permutation_representative()for hyperellitic component
permutation_representative()for non hyperelliptic component, the algorithm is the same as for connected component
for odd component
permutation_representative()for even component
permutation_representative()
The inverse operation (pass from an interval exchange transformation to the connected component) is partially written in [KonZor03] and simply named here ?
Some of the code here was first available on Mathematica [ZS].
A refinement of Zorich representatives was worked out by L. Jefreys in [Jef19]. Namely, for each connected component of Abelian differential his construction provides a square-tiled surface with both in horizontal and vertical direction a decomposition with single cylinder of height one. The implementation is available as
for connected stratum
single_cylinder_representative()for hyperelliptic component
single_cylinder_representative()for odd component
single_cylinder_representative()for even component
single_cylinder_representative()
AUTHORS:
Vincent Delecroix (2009-09-29): initial version
EXAMPLES:
sage: from surface_dynamics import *
Construction of a stratum from a list of singularity degrees:
sage: a = Stratum([1,1], k=1)
sage: a
H_2(1^2)
sage: a.surface_genus()
2
sage: a.dimension()
5
sage: a = Stratum([4,3,2,1], k=1)
sage: a
H_6(4, 3, 2, 1)
sage: a.surface_genus()
6
sage: a.dimension()
15
By convention, the degrees are always written in decreasing order:
sage: a1 = Stratum([4,3,2,1], k=1)
sage: a1
H_6(4, 3, 2, 1)
sage: a2 = Stratum([2,3,1,4], k=1)
sage: a2
H_6(4, 3, 2, 1)
sage: a1 == a2
True
It is possible to lis strata and their connected components:
sage: Stratum([10], k=1).components()
(H_6(10)^hyp, H_6(10)^odd, H_6(10)^even)
Get a list of strata with constraints on genus or on the number of intervals of a representative:
sage: AbelianStrata(genus=3).list()
[H_3(4), H_3(3, 1), H_3(2^2), H_3(2, 1^2), H_3(1^4)]
Obtains the connected components of a stratum:
sage: a = Stratum([0], k=1)
sage: a.components()
(H_1(0)^hyp,)
sage: @cached_function
....: def nb_irred_perm(n):
....: if n == 0 or n == 1: return 1
....: return factorial(n) - sum(nb_irred_perm(k) * factorial(n - k) for k in range(1,n))
sage: [nb_irred_perm(i) for i in range(10)]
[1, 1, 1, 3, 13, 71, 461, 3447, 29093, 273343]
sage: A = AbelianStrata(dimension=5, fake_zeros=True)
sage: N = 0
sage: for a in A:
....: for cc in a.components():
....: for z in set(a.signature()):
....: p = cc.permutation_representative(left_degree=z)
....: n = p.rauzy_diagram().cardinality()
....: print("%13s, %d : %d"%(cc, z, n))
....: print(p)
....: N += n
H_2(2, 0)^hyp, 0 : 11
0 1 2 3 4
4 2 1 3 0
H_2(2, 0)^hyp, 2 : 35
0 1 2 3 4
4 1 3 2 0
H_2(1^2)^hyp, 1 : 15
0 1 2 3 4
4 3 2 1 0
H_1(0^4)^hyp, 0 : 10
0 1 2 3 4
4 0 1 2 3
sage: N
71
sage: nb_irred_perm(5)
71
- surface_dynamics.flat_surfaces.abelian_strata.ASC#
alias of
AbelianStratumComponent
- class surface_dynamics.flat_surfaces.abelian_strata.AbelianStrata(genus=None, dimension=None, fake_zeros=None)#
Abelian strata.
INPUT:
genus- a non negative integer or Nonedimension- a non negative integer or Nonefake_zeros- boolean
EXAMPLES:
sage: from surface_dynamics import *
Abelian strata with a given genus:
sage: for s in AbelianStrata(genus=1): print(s) H_1(0)
sage: for s in AbelianStrata(genus=2): print(s) H_2(2) H_2(1^2)
sage: for s in AbelianStrata(genus=3): print(s) H_3(4) H_3(3, 1) H_3(2^2) H_3(2, 1^2) H_3(1^4)
sage: for s in AbelianStrata(genus=4): print(s) H_4(6) H_4(5, 1) H_4(4, 2) H_4(4, 1^2) H_4(3^2) H_4(3, 2, 1) H_4(3, 1^3) H_4(2^3) H_4(2^2, 1^2) H_4(2, 1^4) H_4(1^6)
Get outside of the tests. Abelian strata with a given number of intervals
sage for s in AbelianStrata(dimension=2): print(s) H^out([0])
sage for s in AbelianStrata(dimension=3): print(s) H^out([0], 0)
sage for s in AbelianStrata(dimension=4): print(s) H^out([2]) H^out([0], 0, 0)
Get outside of tests sage for s in AbelianStrata(dimension=5): print(s) H^out(2, [0]) H^out([2], 0) H^out([1], 1) H^out([0], 0, 0, 0)
- class surface_dynamics.flat_surfaces.abelian_strata.AbelianStrata_all(genus=None, dimension=None, fake_zeros=None)#
Abelian strata.
INPUT:
fake_zeros- boolean (default:False)
EXAMPLES:
sage: from surface_dynamics import * sage: A = AbelianStrata() sage: it = iter(A) sage: for _ in range(10): ....: print(next(it)) H_1(0) H_2(2) H_2(1^2) H_3(4) H_3(3, 1) H_3(2^2) H_4(6) H_3(2, 1^2) H_4(5, 1) H_4(4, 2) sage: A = AbelianStrata(fake_zeros=True) sage: it = iter(A) sage: for _ in range(10): ....: print(next(it)) H_1(0) H_1(0^2) H_2(2) H_1(0^3) H_2(2, 0) H_2(1^2) H_1(0^4) H_3(4) H_2(2, 0^2) H_2(1^2, 0)
- class surface_dynamics.flat_surfaces.abelian_strata.AbelianStrata_d(genus=None, dimension=None, fake_zeros=None)#
Strata with prescribed dimension.
INPUT:
dimension- an integer greater than 1fake_zeros- boolean (default: False) - allows or not fake zeros
EXAMPLES:
sage: from surface_dynamics import * sage: for a in AbelianStrata(dimension=5,fake_zeros=True): ....: print(a) ....: print(a.permutation_representative()) H_2(2, 0) 0 1 2 3 4 4 1 3 2 0 H_2(1^2) 0 1 2 3 4 4 3 2 1 0 H_1(0^4) 0 1 2 3 4 4 0 1 2 3
- an_element()#
Returns the first stratum.
EXAMPLES:
sage: from surface_dynamics import AbelianStrata sage: AbelianStrata(dimension=2).first() H_1(0) sage: AbelianStrata(dimension=3).first() H_1(0^2) sage: AbelianStrata(dimension=4).first() H_2(2)
- cardinality()#
Return the number of Abelian strata with given dimension.
EXAMPLES:
sage: from surface_dynamics import * sage: AbelianStrata(dimension=5,fake_zeros=True).cardinality() 3 sage: AbelianStrata(dimension=5,fake_zeros=False).cardinality() 1 sage: AbelianStrata(dimension=6,fake_zeros=True).cardinality() 4 sage: AbelianStrata(dimension=6,fake_zeros=False).cardinality() 1 sage: AbelianStrata(dimension=7,fake_zeros=True).cardinality() 6 sage: AbelianStrata(dimension=7,fake_zeros=False).cardinality() 2 sage: AbelianStrata(dimension=12,fake_zeros=True).cardinality() 29 sage: AbelianStrata(dimension=12,fake_zeros=False).cardinality() 7
TESTS:
sage: for d in range(1,15): ....: A = AbelianStrata(dimension=d,fake_zeros=True) ....: assert len(A.list()) == A.cardinality() ....: A = AbelianStrata(dimension=d,fake_zeros=False) ....: assert len(A.list()) == A.cardinality()
- first()#
Returns the first stratum.
EXAMPLES:
sage: from surface_dynamics import AbelianStrata sage: AbelianStrata(dimension=2).first() H_1(0) sage: AbelianStrata(dimension=3).first() H_1(0^2) sage: AbelianStrata(dimension=4).first() H_2(2)
- last()#
Return the last stratum.
EXAMPLES:
sage: from surface_dynamics import AbelianStrata sage: AbelianStrata(dimension=9,fake_zeros=True).last() H_1(0^8) sage: AbelianStrata(dimension=9,fake_zeros=False).last() H_3(1^4) sage: AbelianStrata(dimension=10,fake_zeros=True).last() H_1(0^9) sage: AbelianStrata(dimension=10,fake_zeros=False).last() H_4(2^3)
- class surface_dynamics.flat_surfaces.abelian_strata.AbelianStrata_g(genus=None, dimension=None, fake_zeros=None)#
Stratas of genus g surfaces without fake zeros.
INPUT:
genus- a non negative integer
EXAMPLES:
sage: from surface_dynamics import * sage: AbelianStrata(genus=2).list() [H_2(2), H_2(1^2)] sage: AbelianStrata(genus=3).list() [H_3(4), H_3(3, 1), H_3(2^2), H_3(2, 1^2), H_3(1^4)] sage: AbelianStrata(genus=4).random_element() #random H_4(4, 2)
- an_element_()#
Return the first element of this list of strata.
EXAMPLES:
sage: from surface_dynamics import AbelianStrata sage: AbelianStrata(genus=3).first() H_3(4) sage: AbelianStrata(genus=4).first() H_4(6)
- cardinality()#
Return the number of abelian strata with a given genus.
EXAMPLES:
sage: from surface_dynamics import * sage: AbelianStrata(genus=1).cardinality() 1 sage: AbelianStrata(genus=2).cardinality() 2 sage: AbelianStrata(genus=3).cardinality() 5 sage: AbelianStrata(genus=4).cardinality() 11
- first()#
Return the first element of this list of strata.
EXAMPLES:
sage: from surface_dynamics import AbelianStrata sage: AbelianStrata(genus=3).first() H_3(4) sage: AbelianStrata(genus=4).first() H_4(6)
- last()#
Return the last element of this list of strata.
EXAMPLES:
sage: from surface_dynamics import * sage: AbelianStrata(genus=4).last() H_4(1^6) sage: AbelianStrata(genus=5).last() H_5(1^8)
- random_element()#
Return a random stratum.
- class surface_dynamics.flat_surfaces.abelian_strata.AbelianStrata_gd(genus=None, dimension=None, fake_zeros=None)#
Abelian strata of prescribed genus and number of intervals.
INPUT:
genus- integer: the genus of the surfacesdimension- integer: the number of intervalsfake_zeros- boolean: whether or not consider fake zeros
- class surface_dynamics.flat_surfaces.abelian_strata.AbelianStratum(signature, k=1)#
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
surface_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:
sage: from surface_dynamics import *
Creation of an Abelian stratum and get its connected components:
sage: a = Stratum((2, 2), k=1) sage: a H_3(2^2) sage: a.components() (H_3(2^2)^hyp, H_3(2^2)^odd)
Get a permutation representative of a connected component:
sage: a = Stratum((2,2), k=1) sage: a_hyp, a_odd = a.components() sage: a_hyp.permutation_representative() 0 1 2 3 4 5 6 6 5 4 3 2 1 0 sage: a_odd.permutation_representative() 0 1 2 3 4 5 6 3 2 4 6 5 1 0
You can specify the alphabet:
sage: a_odd.permutation_representative(alphabet="ABCDEFGHIJKLMNOPQRSTUVWXYZ") A B C D E F G D C E G F B A
- cylinder_diagram_iterator(ncyls=None, up_to_symmetry=True, force_computation=False)#
Iterator over all cylinder diagram of this stratum.
The generation is up to isomorphism and horizontal/vertical symmetry (and they are in standard form).
INPUT:
ncyls– an optional number of cylindersup_to_symmetry- (boolean, defaultTrue) to return only cylinder diagrams up to horizontal and vertical symmetry.force_computation– ifTruedo no use the database of cylinder diagrams (default isFalse)
EXAMPLES:
sage: from surface_dynamics import Stratum, CylinderDiagram sage: A = Stratum([4], k=1) sage: C1 = [CylinderDiagram('(0,2,1)-(0,3,4) (3)-(2) (4)-(1)'), ....: CylinderDiagram('(0,2,1)-(0,3,4) (3)-(1) (4)-(2)'), ....: CylinderDiagram('(0,1)-(0,3,4) (2,3)-(1) (4)-(2)'), ....: CylinderDiagram('(0,2)-(4) (1,4)-(2,3) (3)-(0,1)'), ....: CylinderDiagram('(0,2)-(0,3) (1,3)-(1,4) (4)-(2)'), ....: CylinderDiagram('(0,1)-(0,3) (2,3)-(1,4) (4)-(2)')] sage: C2 = list(A.cylinder_diagram_iterator(3, force_computation=True)) sage: assert len(C1) == len(C2) sage: for (c1, c2) in zip(C1, C2): ....: assert c1.is_isomorphic(c2) or \ ....: c1.is_isomorphic(c2.horizontal_symmetry()) or \ ....: c1.is_isomorphic(c2.vertical_symmetry()) or \ ....: c1.is_isomorphic(c2.inverse()) sage: sum(1 for _ in A.cylinder_diagram_iterator(3, True, True)) 6 sage: sum(1 for _ in A.cylinder_diagram_iterator(3, True, False)) 6 sage: sum(1 for _ in A.cylinder_diagram_iterator(3, False, True)) 9 sage: sum(1 for _ in A.cylinder_diagram_iterator(3, False, False)) 9
- cylinder_diagrams(ncyls=None, up_to_symmetry=True, force_computation=False)#
Return a list of cylinder diagram of this stratum.
INPUT:
- ``ncyls`` -- an optional number of cylinders
up_to_symmetry- (boolean, defaultTrue) to return only cylinder diagrams up to horizontal and vertical symmetry.force_computation– IfTruethen do not use the database of cylinder diagrams (default isFalse).
EXAMPLES:
sage: from surface_dynamics import Stratum, CylinderDiagram sage: A = Stratum([2,2], k=1) sage: C1 = [CylinderDiagram('(0,1)-(0,5) (2)-(4) (3,4)-(1) (5)-(2,3)'), ....: CylinderDiagram('(0,2,1)-(3,4,5) (3)-(1) (4)-(2) (5)-(0)'), ....: CylinderDiagram('(0,2,1)-(3,5,4) (3)-(1) (4)-(2) (5)-(0)'), ....: CylinderDiagram('(0,3)-(5) (1)-(0) (2,5)-(3,4) (4)-(1,2)'), ....: CylinderDiagram('(0,3)-(0,5) (1,2)-(1,4) (4)-(3) (5)-(2)'), ....: CylinderDiagram('(0,5)-(3,4) (1,4)-(2,5) (2)-(0) (3)-(1)'), ....: CylinderDiagram('(0,5)-(3,4) (1,4)-(2,5) (2)-(1) (3)-(0)')] sage: C2 = A.cylinder_diagrams(4) sage: assert len(C1) == len(C2) sage: isoms = [] sage: for c in A.cylinder_diagrams(4): ....: isom = [] ....: for i,cc in enumerate(C1): ....: if c.is_isomorphic(cc) or \ ....: c.is_isomorphic(cc.horizontal_symmetry()) or \ ....: c.is_isomorphic(cc.vertical_symmetry()) or \ ....: c.is_isomorphic(cc.inverse()): ....: isom.append(i) ....: assert len(isom) == 1, isom ....: isoms.extend(isom) sage: assert sorted(isoms) == [0, 1, 2, 3, 4, 5, 6] sage: len(A.cylinder_diagrams(4, up_to_symmetry=False)) 7 sage: sum(4 / (1 + sum(cd.symmetries())) for cd in A.cylinder_diagrams(4, up_to_symmetry=True)) 7
Recovering the multiplicity of the symmetric versions:
sage: total = 0 sage: for c in Stratum([2,1,1], k=1).cylinder_diagrams(2): ....: total += 4 // (1 + sum(c.symmetries())) sage: total 61 sage: len(Stratum([2, 1, 1], k=1).cylinder_diagrams(2, up_to_symmetry=False)) 61
You obtain the same number directly:
sage: Stratum([2, 1, 1], k=1).cylinder_diagrams_number(2, up_to_symmetry=False) 61
- cylinder_diagrams_by_component(ncyls=None, up_to_symmetry=True, force_computation=False)#
Return a dictionary component -> list of cylinder diagrams.
INPUT:
ncyls- None or integer (default: None) - the number of cylindersup_to_symmetry- (boolean, defaultTrue) to return only cylinder diagrams up to horizontal and vertical symmetry.force_computation- boolean (default:False) - ifFalse, then try to use the database.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: A = Stratum([4], k=1) sage: cyls = A.cylinder_diagrams_by_component(ncyls=2, force_computation=True) sage: A_hyp = A.hyperelliptic_component() sage: A_odd = A.odd_component() sage: len(cyls[A_odd]) 4 sage: len(cyls[A_hyp]) 2 sage: all(c.ncyls() == 2 for c in cyls[A_hyp]) True sage: all(c.stratum_component() == A_hyp for c in cyls[A_hyp]) True sage: all(c.ncyls() == 2 for c in cyls[A_odd]) True sage: all(c.stratum_component() == A_odd for c in cyls[A_odd]) True sage: for ncyls in range(1, 4): ....: for up_to_symmetry in [True, False]: ....: cd1 = A.cylinder_diagrams_by_component(ncyls, up_to_symmetry, True) ....: cd2 = A.cylinder_diagrams_by_component(ncyls, up_to_symmetry, False) ....: assert len(cd1[A_hyp]) == len(cd2[A_hyp]) ....: assert len(cd1[A_odd]) == len(cd2[A_odd])
- cylinder_diagrams_number(ncyls=None, up_to_symmetry=True, force_computation=False)#
Return the number of cylinder diagram that belongs to this stratum.
INPUT:
ncyls– an optional number of cylindersup_to_symmetry- (boolean, defaultTrue) to return only cylinder diagrams up to horizontal and vertical symmetry.force_computation– ifTruedo no use the database of cylinder diagrams (default isFalse)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: H22 = Stratum([2,2], k=1) sage: H22.cylinder_diagrams_number(3) 18 sage: H22.cylinder_diagrams_number(4) 7
If
force_computationis set toTruethen the database is not used. It might be slower for large strata:sage: H22.cylinder_diagrams_number(3, force_computation=True) 18 sage: H22.cylinder_diagrams_number(4, force_computation=True) 7 sage: H31 = Stratum([3,1], k=1) sage: for d in range(1,5): ....: print("%d %d" %(H31.cylinder_diagrams_number(d, True, False), ....: H31.cylinder_diagrams_number(d, True, True))) 2 2 12 12 16 16 4 4 sage: H211 = Stratum([2,1,1], k=1) sage: for d in range(1,6): ....: print("%d %d" % (H211.cylinder_diagrams_number(d, True, False), ....: H211.cylinder_diagrams_number(d, True, True))) 5 5 29 29 53 53 27 27 8 8
- even_component()#
Return the even component of self (if any)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: a = Stratum({2:4}, k=1) sage: a H_5(2^4) sage: a.even_component() H_5(2^4)^even
- has_even_component()#
Test whether this stratum has an even spin component.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: Stratum((2,2), k=1).has_even_component() False sage: Stratum((6,), k=1).has_even_component() True sage: Stratum((6,), k=1).even_component() H_4(6)^even sage: Stratum((0,), k=1).has_even_component() False
- has_hyperelliptic_component()#
Test whether this stratum has an hyperelliptic component.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: Stratum((2,1,1), k=1).has_hyperelliptic_component() False sage: Stratum((2,2), k=1).has_hyperelliptic_component() True sage: Stratum((0,0,0), k=1).has_hyperelliptic_component() True sage: Stratum((2,0), k=1).has_hyperelliptic_component() True sage: Stratum((2,2), k=1).hyperelliptic_component() H_3(2^2)^hyp
- has_non_hyperelliptic_component()#
Test whether this stratum has a non-hyperelliptic component.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: Stratum((1,1), k=1).has_non_hyperelliptic_component() False sage: Stratum((3,3), k=1).has_non_hyperelliptic_component() True sage: Stratum((3,3,0), k=1).has_non_hyperelliptic_component() True sage: Stratum((3,3), k=1).non_hyperelliptic_component() H_4(3^2)^nonhyp
- has_odd_component()#
Test whether this stratum has an odd spin component.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: Stratum((2,), k=1).has_odd_component() False sage: Stratum((4,), k=1).has_odd_component() True sage: Stratum((4,), k=1).odd_component() H_3(4)^odd sage: Stratum((0,), k=1).has_odd_component() False
- hyperelliptic_component()#
Return the hyperelliptic component of self (if any)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: a = Stratum([10], k=1) sage: a H_6(10) sage: a.hyperelliptic_component() H_6(10)^hyp
- non_hyperelliptic_component()#
Return the non hyperelliptic component of self (if any)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: a = Stratum((3,3), k=1) sage: a H_4(3^2) sage: a.non_hyperelliptic_component() H_4(3^2)^nonhyp
- odd_component()#
Return the odd component of self (if any).
EXAMPLES:
sage: from surface_dynamics import Stratum sage: a = Stratum([2,2], k=1) sage: a H_3(2^2) sage: a.odd_component() H_3(2^2)^odd
- one_cylinder_diagram()#
Return a diagram with one cylinder in this connected component.
The diagram returned is the one deduced from the method representative.
INPUT:
ncyls- the number of cylinders
EXAMPLES:
sage: from surface_dynamics import Stratum sage: a = Stratum([3,2,1], k=1) sage: a H_4(3, 2, 1) sage: c = a.one_cylinder_diagram();c (0,8,3,2,1,6,5,4,7)-(0,8,7,6,5,4,3,2,1) sage: c.stratum() H_4(3, 2, 1)
- orientation_quotients(fake_zeros=False)#
Return the list of quadratic strata such that their orientation cover are contained in this stratum.
If
fake_zeros(default: False) is True we do care about poles which becomes a marked zero.EXAMPLES:
sage: from surface_dynamics import Stratum
The stratum H(2g-2) has one conic singularities of angle 2(2g-1)pi. The only way a surface in H(2g-2) covers a quadratic differential is that the quadratic differential has as unique zeros a conical singularity of angle (2g-1) pi. The number of poles may vary and give a collection of possibilities:
sage: Stratum([2], k=1).orientation_quotients() [Q_0(1, -1^5)] sage: Stratum([4], k=1).orientation_quotients() [Q_1(3, -1^3), Q_0(3, -1^7)] sage: Stratum([6], k=1).orientation_quotients() [Q_2(5, -1), Q_1(5, -1^5), Q_0(5, -1^9)]
A stratum with two zeros may or may not have orientation quotients:
sage: Stratum([1,1], k=1).orientation_quotients() [Q_1(2, -1^2), Q_0(2, -1^6)] sage: Stratum([2,2], k=1).orientation_quotients() [Q_1(1^2, -1^2), Q_0(1^2, -1^6), Q_1(4, -1^4), Q_0(4, -1^8)] sage: Stratum([3,1], k=1).orientation_quotients() []
To impose that covering of poles are fake zeros, switch option
fake_zerostoTrue:sage: Stratum([2,2,0,0], k=1).orientation_quotients(fake_zeros=True) [Q_1(1^2, -1^2)]
- separatrix_diagram_iterator(ncyls=None)#
Return an iterator over the separatrix diagrams of this stratum.
For strata of small dimension, it could be faster to use the method separatrix_diagrams.
INPUT:
ncyls– an optional number of cylinders
- separatrix_diagrams(ncyls=None)#
Returns the list of separatrix diagrams that appears in this stratum.
INPUT:
database- boolean (default: True) - if True, use the FlatSurfacesDatabase
EXAMPLES:
sage: from surface_dynamics import Stratum sage: a = Stratum([2], k=1) sage: a H_2(2) sage: for s in a.separatrix_diagrams(): print(s) (0,1,2)-(0,1,2) (0)(1,2)-(0,1)(2)
TESTS:
sage: from surface_dynamics import Stratum sage: for (zeros, ncyl) in [((4,), 3), ((2,2), 4)]: ....: S = Stratum(zeros, k=1).separatrix_diagrams(3) ....: for i in range(len(S)): ....: for j in range(i): ....: assert not S[i].is_isomorphic(S[j])
- separatrix_diagrams_number(ncyls=None)#
Return the number of separatrix diagram that belongs to this stratum.
- single_cylinder_origami()#
Returns an origami associated to a single cylinder permutation representative.
Returns an origami in this connected component having a single vertical cylinder and a single horizontal cylinder.
Examples:
sage: from surface_dynamics import Stratum sage: C = Stratum([4], k=1) sage: O = C.single_cylinder_origami() sage: O (1,2,3,4,5) (1,4,3,5,2) sage: O.stratum() == Stratum([4], k=1) True sage: C = Stratum([2,0], k=1) sage: O = C.single_cylinder_origami() sage: O (1,2,3,4) (1,3,2,4) sage: O.stratum() == Stratum([2], k=1) True
- single_cylinder_representative(alphabet=None, reduced=True)#
Returns a single cylinder permutation representative.
Returns a permutation representative of a square-tiled surface in this component having a single vertical cylinder and a single horizontal cylinder.
Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].
INPUT:
alphabet– an optional alphabet for the permutation representativereduced(boolean, defaultTrue) – whether to return a reduced permutation (ie without labels)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: C = Stratum([2,0], k=1) sage: p = C.single_cylinder_representative() sage: p 0 1 2 3 4 4 3 1 2 0 sage: p.stratum() == C True sage: C = Stratum([3,1], k=1) sage: p = C.single_cylinder_representative(alphabet=Alphabet(name='lower')) sage: p a b c d e f g c f b g e d a sage: p.stratum() == C True sage: C = Stratum([2], k=1) sage: C.single_cylinder_representative() Traceback (most recent call last): ... ValueError: no 1,1-square-tiled surfaces in this stratum try again with H_2(2, 0) sage: C = Stratum([1,1], k=1) sage: C.single_cylinder_representative() Traceback (most recent call last): ... ValueError: no 1,1-square-tiled surfaces in this stratum try again with H_2(1^2, 0^2)
- class surface_dynamics.flat_surfaces.abelian_strata.AbelianStratumComponent(stratum)#
Connected component of Abelian stratum.
Warning
Internal class! Do not use directly!
- arithmetic_teichmueller_curves(n, primitive=False)#
Return the arithmetic Teichmueller curves in that component of stratum.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: A = Stratum([2], k=1).hyperelliptic_component(); A H_2(2)^hyp sage: for i in range(3,10): ....: print("%d %d" % (i,len(A.arithmetic_teichmueller_curves(i)))) 3 1 4 1 5 2 6 1 7 2 8 1 9 2 sage: A = Stratum([1,1], k=1).hyperelliptic_component(); A H_2(1^2)^hyp sage: for i in range(4,10): ....: T = A.arithmetic_teichmueller_curves(i) ....: T_prim = list(filter(lambda t:t.origami().is_primitive(), T)) ....: print("%d %d %d" % (i,len(T),len(T_prim))) 4 2 1 5 1 1 6 5 2 7 2 2 8 4 2 9 4 2 sage: A = Stratum([4], k=1).hyperelliptic_component(); A H_3(4)^hyp sage: for i in range(5,10): ....: print("%d %d" % (i,len(A.arithmetic_teichmueller_curves(i)))) 5 2 6 4 7 3 8 3 9 4
- cylinder_diagram_iterator(ncyls=None, up_to_symmetry=True, force_computation=False)#
An iterator over the cylinder diagrams.
INPUT:
- ``ncyls`` -- (optional) a fixed number of cylinders
up_to_symmetry- (boolean, defaultTrue) to return only cylinder diagrams up to horizontal and vertical symmetry.force_computation– (defaultFalse) whether the database should be used or not
EXAMPLES:
sage: from surface_dynamics import Stratum sage: A = Stratum([1,1,1,1], k=1) sage: cc = A.unique_component() sage: it = cc.cylinder_diagram_iterator(3) sage: cyl = next(it); cyl (0,7)-(0,5) (1,6)-(1,4) (2,4,3,5)-(2,7,3,6) sage: cyl.stratum_component() H_3(1^4)^c sage: cyl.ncyls() 3
Note that if you set
force_computationtoTruethe order of the iteration might be different and you might obtain cylinder diagram with some symmetries applied:sage: # long time sage: C1 = list(cc.cylinder_diagram_iterator(3, force_computation=False)) sage: C2 = list(cc.cylinder_diagram_iterator(3, force_computation=True)) sage: assert len(C1) == len(C2) sage: isoms = [] sage: for c in C1: ....: isom = [] ....: for i,cc in enumerate(C2): ....: if c.is_isomorphic(cc) or \ ....: c.is_isomorphic(cc.horizontal_symmetry()) or \ ....: c.is_isomorphic(cc.vertical_symmetry()) or \ ....: c.is_isomorphic(cc.inverse()): ....: isom.append(i) ....: assert len(isom) == 1, isom ....: isoms.extend(isom) sage: assert sorted(isoms) == list(range(len(C1)))
- cylinder_diagrams(ncyls=None, up_to_symmetry=True, force_computation=False)#
Return the list of cylinder diagrams associated to this component.
INPUT:
ncyls- integer or list of integers (default: None) - consider only the cylinder diagrams with a given number of cylinders.up_to_symmetry- (boolean, defaultTrue) to return only cylinder diagrams up to horizontal and vertical symmetry.force_computation– boolean (defaultFalse). IfTrue, the database of cylinder diagrams is not used.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: C = Stratum([1,1,1,1], k=1).unique_component(); C H_3(1^4)^c sage: for c in C.cylinder_diagrams(6): print(c) (0,1)-(7) (2)-(0) (3)-(4) (4,7)-(5,6) (5)-(1) (6)-(2,3) (0,1)-(7) (2)-(1) (3)-(0) (4,7)-(5,6) (5)-(4) (6)-(2,3) (0,3)-(6,7) (1,2)-(4,5) (4)-(1) (5)-(3) (6)-(2) (7)-(0) (0,3)-(6,7) (1,2)-(4,5) (4)-(3) (5)-(0) (6)-(2) (7)-(1)
- cylinder_diagrams_number(ncyls=None, up_to_symmetry=True, force_computation=False)#
Return the number of cylinder diagrams.
INPUT:
ncyls- integer or list of integers (default: None) - restrict the counting to a given number of cylinders.up_to_symmetry- (boolean, defaultTrue) to count only cylinder diagrams up to horizontal and vertical symmetry.force_computation- (default:False) whether we use the database or compute explicitly using the generation algorithm.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: C = Stratum([3,1], k=1).unique_component() sage: C.cylinder_diagrams_number(1) 2 sage: C.cylinder_diagrams_number(2) 12 sage: C.cylinder_diagrams_number(3) 16 sage: C.cylinder_diagrams_number(4) 4
Note that when setting
force_computationtoTruewe got the same numbers:sage: for i in range(1,5): ....: print(C.cylinder_diagrams_number(i, force_computation=True)) 2 12 16 4 sage: C = Stratum([6], k=1) sage: C_hyp = C.hyperelliptic_component() sage: C_odd = C.odd_component() sage: C_even = C.even_component() sage: for i in range(1,5): print(C.cylinder_diagrams_number(i)) 16 76 130 67 sage: for i in range(1,5): print(C_hyp.cylinder_diagrams_number(i)) 1 3 8 4 sage: for i in range(1,5): print(C_odd.cylinder_diagrams_number(i)) 11 49 80 42 sage: for i in range(1,5): ....: print(C_even.cylinder_diagrams_number(i, True, False)) 4 24 42 21 sage: for i in range(1,5): # long time ....: print(C_even.cylinder_diagrams_number(i, True, True)) # long time 4 24 42 21
- lyapunov_exponents(**kargs)#
Return the approximate Lyapunov exponents of the KZ-cocycle.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: Stratum([2], k=1).unique_component().lyapunov_exponents_approx(nb_iterations=2**21) # abs tol .05 [1.000, 0.333] sage: H4hyp, H4odd = Stratum([4], k=1).components() sage: H4hyp.lyapunov_exponents_approx(nb_iterations=2**21) # abs tol .05 [1.000, 0.616, 0.184] sage: H4odd.lyapunov_exponents_approx(nb_iterations=2**21) # abs tol .05 [1.000, 0.418, 0.182]
- lyapunov_exponents_approx(**kargs)#
Return the approximate Lyapunov exponents of the KZ-cocycle.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: Stratum([2], k=1).unique_component().lyapunov_exponents_approx(nb_iterations=2**21) # abs tol .05 [1.000, 0.333] sage: H4hyp, H4odd = Stratum([4], k=1).components() sage: H4hyp.lyapunov_exponents_approx(nb_iterations=2**21) # abs tol .05 [1.000, 0.616, 0.184] sage: H4odd.lyapunov_exponents_approx(nb_iterations=2**21) # abs tol .05 [1.000, 0.418, 0.182]
- one_cylinder_diagram()#
Return a diagram with one cylinder in this connected component.
The diagram returned is the one deduced from the method permutation_representative.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: A = Stratum([2,2], k=1).odd_component() sage: c = A.one_cylinder_diagram() sage: c (0,5,1,3,2,4)-(0,5,4,3,2,1) sage: c.stratum_component() H_3(2^2)^odd sage: A = Stratum([3,3], k=1).non_hyperelliptic_component() sage: c = A.one_cylinder_diagram() sage: c (0,7,3,2,1,5,4,6)-(0,7,6,5,4,3,2,1) sage: c.stratum_component() H_4(3^2)^nonhyp
- one_origami()#
Returns an origami in this component
The origami returned has the minimal number of squares and one cylinder. It is obtained from the permutation representative of the stratum.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: a = Stratum([2,2], k=1).one_component() sage: a.one_origami().stratum() H_3(2^2) sage: Stratum([3,2,1], k=1).unique_component().one_origami().stratum() H_4(3, 2, 1)
- origami_iterator(n, reduced=True, primitive=False)#
Iterator through the set of origamis with
nsquares in this stratum.The output origamis are in normal form. But be careful as there may be repetition in the output!
INPUT:
n- integer - the number of squaresreduced- boolean (default:True)primitive- boolean (default:False)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: cc = Stratum([6], k=1).even_component() sage: it = cc.origami_iterator(13) sage: o = next(it) sage: o (1,2,3,4,5,6,7,8,9,10,11,12,13) (1,8,2)(3,12,6,11,5,13,7,9)(4,10) sage: o.stratum_component() H_4(6)^even
- origamis(n, reduced=True, primitive=False)#
Return the set of origamis with n squares in this stratum.
INPUT:
n- integer - the number of squaresreduced- boolean (default: True)primitive- boolean (default: False)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: H11_hyp = Stratum([1,1], k=1).hyperelliptic_component() sage: len(H11_hyp.origamis(6)) 88 sage: T6 = H11_hyp.arithmetic_teichmueller_curves(6) sage: len(T6) 5 sage: sum(t.veech_group().index() for t in T6) 88 sage: H4_odd = Stratum([4], k=1).odd_component() sage: len(H4_odd.origamis(6)) 155 sage: T6 = H4_odd.arithmetic_teichmueller_curves(6) sage: sum(t.veech_group().index() for t in T6) 155
- permutation_representative(left_degree=None, reduced=True, alphabet=None, relabel=True)#
Returns the Zorich representative of this connected component.
Zorich constructs explicitly 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 permutationleft_degree- the degree of the singularity on the left of the interval.
OUTPUT:
permutation – a permutation which lives in this component
EXAMPLES:
sage: from surface_dynamics import Stratum sage: c = Stratum([1,1,1,1], k=1).unique_component() sage: p = c.permutation_representative(alphabet="abcdefghi") sage: p a b c d e f g h i e d c f i h g b a sage: p.stratum_component() H_3(1^4)^c sage: cc = Stratum([3,2,1,0], k=1).unique_component() sage: p = cc.permutation_representative(left_degree=3); p 0 1 2 3 4 5 6 7 8 9 10 4 3 7 6 5 10 9 8 2 0 1 sage: p.stratum_component() H_4(3, 2, 1, 0)^c sage: p.marking().left() 4 sage: p.rauzy_diagram() # not tested Rauzy diagram with 1060774 permutations sage: p = cc.permutation_representative(left_degree=2); p 0 1 2 3 4 5 6 7 8 9 10 4 3 5 7 6 10 9 8 2 0 1 sage: p.stratum_component() H_4(3, 2, 1, 0)^c sage: p.marking().left() 3 sage: p.rauzy_diagram() # not tested Rauzy diagram with 792066 permutations sage: p = cc.permutation_representative(left_degree=1); p 0 1 2 3 4 5 6 7 8 9 10 5 4 3 7 6 8 10 9 2 0 1 sage: p.stratum_component() H_4(3, 2, 1, 0)^c sage: p.marking().left() 2 sage: p.rauzy_diagram() # not tested Rauzy diagram with 538494 permutations sage: p = cc.permutation_representative(left_degree=0); p 0 1 2 3 4 5 6 7 8 9 10 4 2 7 6 5 10 9 8 1 3 0 sage: p.stratum_component() H_4(3, 2, 1, 0)^c sage: p.marking().left() 1 sage: p.rauzy_diagram() # not tested Rauzy diagram with 246914 permutations
- random_standard_permutation(nsteps=64)#
Perform a random walk on rauzy diagram stopped on a standard permutation.
INPUT:
nsteps- integer or None - perform nsteps and then stops as soon as a Strebel differential is found.
At each step, with probability 1/3 we perform one of the following moves:
exchange top,bottom and left,right (proba 1/10)
top rauzy move (proba 9/20)
bot rauzy move (proba 9/20)
EXAMPLES:
sage: from surface_dynamics import Stratum
sage: C = Stratum([10], k=1).hyperelliptic_component() sage: p = C.random_standard_permutation(); p # random 0 1 2 3 4 5 6 7 8 9 10 11 11 10 9 8 7 6 5 4 3 2 1 0 sage: p.stratum_component() H_6(10)^hyp
sage: C = Stratum([6,4,2], k=1).odd_component(); C H_7(6, 4, 2)^odd sage: p = C.random_standard_permutation(); p # random 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 15 2 14 12 3 11 6 10 8 5 9 13 7 4 1 0 sage: p.stratum_component() H_7(6, 4, 2)^odd
sage: C = Stratum([2,2,2,2], k=1).even_component(); C H_5(2^4)^even sage: p = C.random_standard_permutation(); p # random 0 1 2 3 4 5 6 7 8 9 10 11 12 12 4 9 11 8 3 7 6 1 10 2 5 0 sage: p.stratum_component() H_5(2^4)^even
sage: C = Stratum([32], k=1).odd_component(); C H_17(32)^odd sage: p = C.random_standard_permutation(); p # random 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 33 30 10 3 32 19 11 28 4 14 24 15 21 20 9 12 25 6 2 29 26 23 27 13 8 1 18 17 16 31 7 22 5 0 sage: p.stratum_component() H_17(32)^odd
- rauzy_class_cardinality(left_degree=None, reduced=True)#
Rauzy diagram cardinality for connected components.
Returns the cardinality of the extended Rauzy diagram associated to this connected component.
If left_degree is provided then it returns the cardinality of the Rauzy diagram with a singularity of that degree attached on the left. Otherwise it returns the cardinality of the extended Rauzy diagram.
INPUT:
left_degree- the degree to be attached to the singularity on the leftreduced- boolean (default: True) - consider the cardinality of reduced or extended Rauzy diagram
EXAMPLES:
sage: from surface_dynamics import * sage: a = Stratum({1:4}, k=1).unique_component() sage: a H_3(1^4)^c sage: a.rauzy_diagram() Rauzy diagram with 1255 permutations sage: a.rauzy_class_cardinality() 1255 sage: cc = Stratum([3,2,1], k=1).unique_component() sage: cc.rauzy_diagram(left_degree=3) # long time Rauzy diagram with 96434 permutations sage: cc.rauzy_class_cardinality(left_degree=3) 96434 sage: cc.rauzy_diagram(left_degree=2) # long time Rauzy diagram with 72006 permutations sage: cc.rauzy_class_cardinality(left_degree=2) 72006 sage: cc.rauzy_diagram(left_degree=1) # long time Rauzy diagram with 48954 permutations sage: cc.rauzy_class_cardinality(left_degree=1) 48954 sage: a = Stratum({1:8}, k=1).unique_component() sage: a H_5(1^8)^c sage: a.rauzy_class_cardinality() 55184875
Cardinalities for labeled Rauzy classes instead of reduced:
sage: cc = Stratum([2,1,1], k=1).unique_component() sage: cc.rauzy_diagram(left_degree=2, reduced=False) Rauzy diagram with 3676 permutations sage: cc.rauzy_class_cardinality(left_degree=2, reduced=False) 3676 sage: cc.rauzy_diagram(left_degree=1, reduced=False) Rauzy diagram with 3774 permutations sage: cc.rauzy_class_cardinality(left_degree=1,reduced=False) 3774 sage: cc = Stratum([2,1,1,0], k=1).unique_component() sage: cc.rauzy_diagram(left_degree=2, reduced=False) # long time Rauzy diagram with 33084 permutations sage: cc.rauzy_diagram(left_degree=1, reduced=False) # long time Rauzy diagram with 33966 permutations sage: cc.rauzy_diagram(left_degree=0, reduced=False) # long time Rauzy diagram with 30828 permutations sage: cc.rauzy_class_cardinality(left_degree=2, reduced=False) 33084 sage: cc.rauzy_class_cardinality(left_degree=1, reduced=False) 33966 sage: cc.rauzy_class_cardinality(left_degree=0, reduced=False) 30828
- rauzy_diagram(*args, **kwds)#
Returns the extended Rauzy diagram associated to this connected component.
OUTPUT:
rauzy diagram – the Rauzy diagram associated to this stratum
EXAMPLES:
sage: from surface_dynamics import Stratum sage: c = Stratum([0], k=1).components()[0] sage: r = c.rauzy_diagram()
- single_cylinder_origami()#
Returns an origami associated to a single cylinder permutation representative.
Returns an origami in this connected (or non-hyperelliptic) component having a single vertical cylinder and a single horizontal cylinder.
Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].
Examples:
sage: from surface_dynamics import Stratum sage: cc = Stratum([4], k=1).odd_component() sage: O = cc.single_cylinder_origami() sage: O (1,2,3,4,5) (1,4,3,5,2) sage: O.stratum_component() == cc True sage: cc = Stratum([5,3], k=1).unique_component() sage: O = cc.single_cylinder_origami() sage: O (1,2,3,4,5,6,7,8,9,10) (1,9,8,10,6,7,4,3,5,2) sage: O.stratum_component() == cc True sage: cc = Stratum([4,2], k=1).even_component() sage: O = cc.single_cylinder_origami() sage: O (1,2,3,4,5,6,7,8) (1,3,7,5,6,8,4,2) sage: O.stratum_component() == cc True
- single_cylinder_representative(alphabet=None, reduced=True)#
Returns a single cylinder permutation representative.
Returns a cylindric permutation representative of this connected stratum (or non-hyperelliptic component) such that the associated square-tiled surface is made of a single cylinder of height one in both horizontal and vertical direction.
Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].
INPUT:
alphabet– an optional alphabet for the permutation representativereduced(boolean, defaultTrue) – whether to return a reduced permutation (ie without labels)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: cc = Stratum([1,1,1,1], k=1).unique_component() sage: p = cc.single_cylinder_representative() sage: p 0 1 2 3 4 5 6 7 8 2 6 5 3 1 8 4 7 0 sage: p.stratum_component() == cc True sage: cc = Stratum([2,1,1], k=1).unique_component() sage: p = cc.single_cylinder_representative() sage: p 0 1 2 3 4 5 6 7 2 6 4 1 7 5 3 0 sage: p.stratum_component() == cc True sage: cc = Stratum([3,3], k=1).non_hyperelliptic_component() sage: p = cc.single_cylinder_representative(alphabet=Alphabet(name='lower')) sage: p a b c d e f g h i c i g f h e b d a sage: p.stratum_component() == cc True
- spin()#
Return
Nonesince surfaces in this component have no spin.EXAMPLES:
sage: from surface_dynamics import Stratum sage: c = Stratum([1,1,1,1], k=1).unique_component(); c H_3(1^4)^c sage: c.spin() is None True
- standard_permutations()#
Return the set of standard permutations.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: C = Stratum([4], k=1).odd_component() sage: C H_3(4)^odd sage: for p in C.standard_permutations(): print("%s\n***********" % p) 0 1 2 3 4 5 5 2 1 4 3 0 *********** 0 1 2 3 4 5 5 3 1 4 2 0 *********** 0 1 2 3 4 5 5 4 1 3 2 0 *********** 0 1 2 3 4 5 5 2 4 1 3 0 *********** 0 1 2 3 4 5 5 4 2 1 3 0 *********** 0 1 2 3 4 5 5 2 4 3 1 0 *********** 0 1 2 3 4 5 5 3 2 4 1 0 ***********
- standard_permutations_number(left_degree=None)#
Return the number of standard permutations in the Rauzy class associated to this connected component.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: cc = Stratum([3,1], k=1).unique_component() sage: sum(1 for p in cc.rauzy_diagram() if p.is_standard()) 24 sage: cc.standard_permutations_number() 24 sage: sum(1 for p in cc.rauzy_diagram(left_degree=3) if p.is_standard()) 16 sage: cc.standard_permutations_number(left_degree=3) 16 sage: sum(1 for p in cc.rauzy_diagram(left_degree=1) if p.is_standard()) 8 sage: cc.standard_permutations_number(left_degree=1) 8 sage: cc = Stratum({1:10}, k=1).unique_component() sage: cc H_6(1^10)^c sage: cc.standard_permutations_number() 59520825
- surface_dynamics.flat_surfaces.abelian_strata.DeprecatedAbelianStratumConstructor(*l)#
TESTS:
sage: from surface_dynamics import AbelianStratum, Stratum sage: s = AbelianStratum(0) doctest:warning ... UserWarning: AbelianStratum has changed its arguments in order to handle meromorphic and higher-order differentials; use Stratum instead sage: s is loads(dumps(s)) True sage: AbelianStratum(1,1,1,1) is AbelianStratum(1,1,1,1) True sage: AbelianStratum(1,1,1,1) is Stratum((1, 1, 1, 1), 1) True
- surface_dynamics.flat_surfaces.abelian_strata.EvenASC#
alias of
EvenAbelianStratumComponent
- class surface_dynamics.flat_surfaces.abelian_strata.EvenAbelianStratumComponent(stratum)#
Connected component of Abelian stratum with even spin structure.
Warning
Internal class! Do not use directly!
- permutation_representative(left_degree=None, reduced=True, alphabet=None, relabel=True)#
Returns the Zorich representative of this connected component.
Zorich constructs explicitly interval exchange transformations for each stratum in [Zor08].
INPUT:
reduced- boolean (default: True): whether you obtain a reduced or labelled permutationleft_degree- integer (optional) - a specified degree of zero at the left of the interval.alphabet- alphabet or None (default: None): whether you want to specify an alphabet for your representativerelabel- boolean (default: True) - if False uses Zorich’s natural numbering otherwise uses 0,1,…
EXAMPLES:
sage: from surface_dynamics import Stratum sage: c = Stratum([6], k=1).even_component() sage: c H_4(6)^even sage: p = c.permutation_representative(alphabet=range(8)) sage: p 0 1 2 3 4 5 6 7 5 4 3 2 7 6 1 0 sage: p.stratum_component() H_4(6)^even
sage: c = Stratum([4,4], k=1).even_component() sage: c H_5(4^2)^even sage: p = c.permutation_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.stratum_component() H_5(4^2)^even
Different markings lead to different Rauzy diagrams:
sage: c = Stratum([4,2,0], k=1).even_component() sage: p = c.permutation_representative(left_degree=4); p 0 1 2 3 4 5 6 7 8 9 6 5 4 3 7 9 8 2 0 1 sage: p.stratum_component() H_4(4, 2, 0)^even sage: p.marking().left() 5 sage: p.rauzy_diagram() # long time Rauzy diagram with 66140 permutations sage: p = c.permutation_representative(left_degree=2); p 0 1 2 3 4 5 6 7 8 9 7 6 5 4 3 9 8 2 0 1 sage: p.stratum_component() H_4(4, 2, 0)^even sage: p.marking().left() 3 sage: p.rauzy_diagram() # long time Rauzy diagram with 39540 permutations sage: p = c.permutation_representative(left_degree=0); p 0 1 2 3 4 5 6 7 8 9 6 4 3 2 7 9 8 1 5 0 sage: p.stratum_component() H_4(4, 2, 0)^even sage: p.marking().left() 1 sage: p.rauzy_diagram() # long time Rauzy diagram with 11792 permutations
- rauzy_class_cardinality(left_degree=None, reduced=True)#
Cardinality of rauzy diagram for even component of a stratum
INPUT:
left_degree- integerreduced- boolean
EXAMPLES:
sage: from surface_dynamics import Stratum sage: c = Stratum([6], k=1).even_component() sage: c.rauzy_diagram() Rauzy diagram with 2327 permutations sage: c.rauzy_class_cardinality() 2327 sage: c = Stratum([4,2,0], k=1).even_component() sage: c.rauzy_class_cardinality() 117472 sage: c.rauzy_diagram(left_degree=4) # long time Rauzy diagram with 66140 permutations sage: c.rauzy_class_cardinality(left_degree=4) 66140 sage: c.rauzy_diagram(left_degree=4, reduced=False) # long time Rauzy diagram with 198420 permutations sage: c.rauzy_class_cardinality(left_degree=4,reduced=False) 198420 sage: c.rauzy_class_cardinality(2) 39540 sage: c.rauzy_diagram(left_degree=2) # long time Rauzy diagram with 39540 permutations sage: c.rauzy_diagram(left_degree=2, reduced=False) # long time Rauzy diagram with 197700 permutations sage: c.rauzy_class_cardinality(left_degree=2, reduced=False) 197700 sage: c.rauzy_class_cardinality(0) 11792 sage: c.rauzy_diagram(left_degree=0) Rauzy diagram with 11792 permutations sage: c.rauzy_diagram(left_degree=0, reduced=False) # long time Rauzy diagram with 176880 permutations sage: c.rauzy_class_cardinality(left_degree=0, reduced=False) 176880
- single_cylinder_representative(alphabet=None, reduced=False)#
Returns a single cylinder permutation representative.
Returns a permutation representative of a square-tiled surface in this component having a single vertical cylinder and a single horizontal cylinder.
Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].
INPUT:
alphabet– an optional alphabet for the permutation representativereduced(boolean, defaultTrue) – whether to return a reduced permutation (ie without labels)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: cc = Stratum([6], k=1).even_component() sage: p = cc.single_cylinder_representative(alphabet=Alphabet(name='lower')) sage: p a b c d e f g h c h g f d b e a sage: p.stratum_component() == cc True sage: cc = Stratum([4,4], k=1).even_component() sage: p = cc.single_cylinder_representative() sage: p 0 1 2 3 4 5 6 7 8 9 10 2 10 7 5 8 1 9 6 4 3 0 sage: p.stratum_component() == cc True
- spin()#
Return
0.EXAMPLES:
sage: from surface_dynamics import Stratum sage: c = Stratum([4,2], k=1).even_component(); c H_4(4, 2)^even sage: c.spin() 0
- standard_permutations_number()#
Return the number of standard permutation of this even component.
EXAMPLES:
sage: from surface_dynamics import Stratum
For strata in genus 3, the number of standard permutations is reasonably small and the whole set can be computed:
sage: C = Stratum([6], k=1).even_component() sage: len(C.standard_permutations()) # long time 44 sage: C.standard_permutations_number() 44 sage: C = Stratum([4,2], k=1).even_component() sage: len(C.standard_permutations()) # long time 136 sage: C.standard_permutations_number() 136 sage: C = Stratum([2,2,2], k=1).even_component() sage: len(C.standard_permutations()) # long time 92 sage: C.standard_permutations_number() 92
For higher genera, this number can be very big:
sage: C = Stratum([20], k=1).even_component() sage: C.standard_permutations_number() 109398514483439999
- surface_dynamics.flat_surfaces.abelian_strata.HypASC#
alias of
HypAbelianStratumComponent
- class surface_dynamics.flat_surfaces.abelian_strata.HypAbelianStratumComponent(stratum)#
Hyperelliptic component of Abelian stratum.
- permutation_representative(left_degree=None, reduced=True, alphabet=None, relabel=True)#
Returns the Zorich representative of this connected component.
Zorich constructs explicitly 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: from surface_dynamics import Stratum sage: c = Stratum([0], k=1).hyperelliptic_component() sage: p = c.permutation_representative() sage: p 0 1 1 0 sage: p.stratum_component() H_1(0)^hyp sage: c = Stratum([0,0], k=1).hyperelliptic_component() sage: p = c.permutation_representative(alphabet="abc") sage: p a b c c b a sage: p.stratum_component() H_1(0^2)^hyp sage: c = Stratum([2,2], k=1).hyperelliptic_component() sage: p = c.permutation_representative(alphabet="ABCDEFGHIJKL") sage: p A B C D E F G G F E D C B A sage: c = Stratum([1,1,0], k=1).hyperelliptic_component() sage: p = c.permutation_representative(left_degree=1); p 0 1 2 3 4 5 5 1 4 3 2 0 sage: p.marking().left() 2 sage: p.rauzy_diagram() Rauzy diagram with 90 permutations sage: p = c.permutation_representative(left_degree=0); p 0 1 2 3 4 5 5 3 2 1 4 0 sage: p.marking().left() 1 sage: p.rauzy_diagram() Rauzy diagram with 20 permutations
- random_standard_permutation(nsteps=None)#
In hyperelliptic component there is only one standard permutation.
- rauzy_class_cardinality(left_degree=None, reduced=True)#
Return the cardinality of the extended Rauzy diagram associated to the hyperelliptic component
The cardinality of the Rauzy diagram or extended Rauzy diagram associated to H_{hyp}(2g-2,0^k) or H_{hyp}(g-1,g-1,0^k) depends only on the dimension d of the initial stratum mathcal{H}_{hyp}(2g-2) for which d=2g or mathcal{H}_{hyp}(g-1,g-1) for which d=2g+1 and the number of fake zeros k. The formula is
\[\binom{d+k+1}{k} (2^{d-1}-1) + d \binom{d+k}{k-1}\]INPUT:
left_degree- integer - the degree of the singularity attached at the left of the interval.reduced- boolean (default: True) - if False, consider labeled Rauzy diagrams instead of reduced.
EXAMPLES:
sage: from surface_dynamics import Stratum
The case of the torus is a little bit different:
sage: c = Stratum([0], k=1).hyperelliptic_component() sage: c.rauzy_diagram() Rauzy diagram with 1 permutation sage: c.rauzy_class_cardinality() 1 sage: c = Stratum([0,0], k=1).hyperelliptic_component() sage: c.rauzy_diagram() Rauzy diagram with 3 permutations sage: c.rauzy_class_cardinality() 3
Examples in genus 2:
sage: c = Stratum([2,0], k=1).hyperelliptic_component() sage: c.rauzy_diagram() Rauzy diagram with 46 permutations sage: c.rauzy_class_cardinality() 46 sage: c.rauzy_diagram(left_degree=2) Rauzy diagram with 35 permutations sage: c.rauzy_class_cardinality(left_degree=2) 35 sage: c.rauzy_diagram(left_degree=0) Rauzy diagram with 11 permutations sage: c.rauzy_class_cardinality(left_degree=0) 11 sage: c.rauzy_diagram(left_degree=0, reduced=False) Rauzy diagram with 33 permutations sage: c.rauzy_class_cardinality(left_degree=0, reduced=False) 33 sage: c = Stratum([1,1,0,0], k=1).hyperelliptic_component() sage: c.rauzy_diagram() Rauzy diagram with 455 permutations sage: c.rauzy_class_cardinality() 455 sage: c.rauzy_diagram(left_degree=1) Rauzy diagram with 315 permutations sage: c.rauzy_class_cardinality(left_degree=1) 315 sage: c.rauzy_diagram(left_degree=1, reduced=False) Rauzy diagram with 630 permutations sage: c.rauzy_class_cardinality(left_degree=1, reduced=False) 630 sage: c.rauzy_diagram(left_degree=0) Rauzy diagram with 140 permutations sage: c.rauzy_class_cardinality(left_degree=0) 140 sage: c.rauzy_diagram(left_degree=0, reduced=False) Rauzy diagram with 560 permutations sage: c.rauzy_class_cardinality(left_degree=0, reduced=False) 560
Other examples in higher genus:
sage: c = Stratum([12,0,0], k=1).hyperelliptic_component() sage: c.rauzy_class_cardinality() 1114200 sage: c.rauzy_class_cardinality(left_degree=12, reduced=False) 1965840 sage: c = Stratum([14], k=1).hyperelliptic_component() sage: c.rauzy_class_cardinality() 32767
- single_cylinder_representative(alphabet=None, reduced=True)#
Returns a single cylinder permutation representative.
Returns a permutation representative of a square-tiled surface in this component having a single vertical cylinder and a single horizontal cylinder.
Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].
INPUT:
alphabet– an optional alphabet for the permutation representativereduced(boolean, defaultTrue) – whether to return a reduced permutation (ie without labels)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: cc = Stratum([2,0], k=1).hyperelliptic_component() sage: p = cc.single_cylinder_representative(alphabet=Alphabet(name='upper')) sage: p A B C D E E D B C A sage: p.stratum_component() == cc True sage: cc = Stratum({3:2,0:6}, k=1).hyperelliptic_component() sage: p = cc.single_cylinder_representative() sage: p 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 14 12 13 10 11 8 9 7 5 6 3 4 1 2 0 sage: p.stratum_component() == cc True sage: cc = Stratum([2], k=1).hyperelliptic_component() sage: cc.single_cylinder_representative() Traceback (most recent call last): ... ValueError: no 1,1-square-tiled surfaces in this connected component try again with H_2(2, 0)^hyp sage: cc = Stratum({3:2,0:5}, k=1).hyperelliptic_component() sage: cc.single_cylinder_representative() Traceback (most recent call last): ... ValueError: no 1,1-square-tiled surfaces in this connected component try again with H_4(3^2, 0^6)^hyp
- spin()#
Return the spin parity of hyperelliptic stratum.
EXAMPLES:
sage: from surface_dynamics import Stratum
For the strata H(2g-2):
sage: c = Stratum([0], k=1).hyperelliptic_component() sage: c.spin() 1 sage: p = c.permutation_representative() sage: p.arf_invariant() 1 sage: c = Stratum([2], k=1).hyperelliptic_component() sage: c.spin() 1 sage: p = c.permutation_representative() sage: p.arf_invariant() 1 sage: c = Stratum([4], k=1).hyperelliptic_component() sage: c.spin() 0 sage: p = c.permutation_representative() sage: p.arf_invariant() 0
For the strata H(g-1,g-1):
sage: c = Stratum([2,2], k=1).hyperelliptic_component() sage: c.spin() 0 sage: p = c.permutation_representative() sage: p.arf_invariant() 0 sage: c = Stratum([4,4], k=1).hyperelliptic_component() sage: c.spin() 1 sage: p = c.permutation_representative() sage: p.arf_invariant() 1
- standard_permutations(reduced=True)#
Return the standard permutations in this hyperelliptic component.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: Stratum([6], k=1).hyperelliptic_component().standard_permutations() [0 1 2 3 4 5 6 7 7 6 5 4 3 2 1 0]
- standard_permutations_number()#
Return the number of standard permutations in this hyperelliptic component.
- surface_dynamics.flat_surfaces.abelian_strata.NonHypASC#
alias of
NonHypAbelianStratumComponent
- class surface_dynamics.flat_surfaces.abelian_strata.NonHypAbelianStratumComponent(stratum)#
Non hyperelliptic component of Abelian stratum.
- rauzy_class_cardinality(left_degree=None, reduced=True)#
Return the cardinality of Rauzy diagram associated to this non hyperelliptic component.
INPUT:
left_degree- integerreduced- boolean (default: True)
EXAMPLES:
sage: from surface_dynamics import Stratum
Examples in genus 3:
sage: c = Stratum([3,3], k=1).non_hyperelliptic_component() sage: c.rauzy_class_cardinality() 15568 sage: c = Stratum([3,3,0], k=1).non_hyperelliptic_component() sage: c.rauzy_class_cardinality() 173723 sage: c.rauzy_diagram(left_degree=3) # long time Rauzy diagram with 155680 permutations sage: c.rauzy_class_cardinality(left_degree=3) 155680 sage: c.rauzy_diagram(left_degree=3, reduced=False) # not tested Rauzy diagram with 311360 permutations sage: c.rauzy_class_cardinality(left_degree=3, reduced=False) 311360 sage: c.rauzy_diagram(left_degree=0) # long time Rauzy diagram with 18043 permutations sage: c.rauzy_class_cardinality(left_degree=0) 18043 sage: cc.rauzy_diagram(left_degree=0, reduced=False) # not tested Rauzy diagram with 288688 permutations sage: c.rauzy_class_cardinality(left_degree=0,reduced=False) 288688
When genus growths, the size of the Rauzy diagram becomes very big:
sage: c = Stratum([5,5], k=1).non_hyperelliptic_component() sage: c.rauzy_class_cardinality() 136116680 sage: c = Stratum([7,7,0], k=1).non_hyperelliptic_component() sage: c.rauzy_class_cardinality() 88484743236111 sage: c.rauzy_class_cardinality(left_degree=7, reduced=False) 334071852804864
- standard_permutations_number()#
EXAMPLES:
sage: from surface_dynamics import Stratum sage: C = Stratum([3,3], k=1).non_hyperelliptic_component() sage: len(C.standard_permutations()) # long time 275 sage: C.standard_permutations_number() 275 sage: C = Stratum([5,5], k=1).non_hyperelliptic_component() sage: C.standard_permutations_number() 1022399 sage: C = Stratum([7,7], k=1).non_hyperelliptic_component() sage: C.standard_permutations_number() 19229011199
- surface_dynamics.flat_surfaces.abelian_strata.OddASC#
alias of
OddAbelianStratumComponent
- class surface_dynamics.flat_surfaces.abelian_strata.OddAbelianStratumComponent(stratum)#
Connected component of an Abelian stratum with odd spin parity.
- permutation_representative(left_degree=None, reduced=True, alphabet=None, relabel=True)#
Returns the Zorich representative of this connected component.
A. Zorich constructs explicitly interval exchange transformations for each stratum in [Zor08].
EXAMPLES:
sage: from surface_dynamics import Stratum sage: a = Stratum([6], k=1).odd_component() sage: p = a.permutation_representative() sage: p 0 1 2 3 4 5 6 7 3 2 5 4 7 6 1 0 sage: p.stratum_component() H_4(6)^odd
sage: a = Stratum([4,4], k=1).odd_component() sage: p = a.permutation_representative() sage: p 0 1 2 3 4 5 6 7 8 9 10 3 2 5 4 6 8 7 10 9 1 0 sage: p.stratum_component() H_5(4^2)^odd
Different markings lead to different Rauzy diagrams:
sage: c = Stratum([4,2,0], k=1).odd_component() sage: p = c.permutation_representative(left_degree=4); p 0 1 2 3 4 5 6 7 8 9 4 3 6 5 7 9 8 2 0 1 sage: p.stratum_component() H_4(4, 2, 0)^odd sage: p.marking().left() 5 sage: p.rauzy_diagram() # not tested Rauzy diagram with 147090 permutations sage: p = c.permutation_representative(left_degree=2); p 0 1 2 3 4 5 6 7 8 9 4 3 5 7 6 9 8 2 0 1 sage: p.stratum_component() H_4(4, 2, 0)^odd sage: p.marking().left() 3 sage: p.rauzy_diagram() # long time Rauzy diagram with 87970 permutations sage: p = c.permutation_representative(left_degree=0); p 0 1 2 3 4 5 6 7 8 9 4 2 6 5 7 9 8 1 3 0 sage: p.stratum_component() H_4(4, 2, 0)^odd sage: p.marking().left() 1 sage: p.rauzy_diagram() # long time Rauzy diagram with 27754 permutations
- rauzy_class_cardinality(left_degree=None, reduced=True)#
Cardinality of rauzy diagram for odd component
INPUT:
left_degree- integer (optional)reduced- boolean (default: True)
EXAMPLES:
sage: from surface_dynamics import Stratum
The genus must be at least 3 to have an odd component:
sage: c = Stratum([4], k=1).odd_component() sage: c.rauzy_diagram() Rauzy diagram with 134 permutations sage: c.rauzy_class_cardinality() 134 sage: c = Stratum([4,0], k=1).odd_component() sage: c.rauzy_diagram() Rauzy diagram with 1114 permutations sage: c.rauzy_class_cardinality() 1114 sage: c = Stratum([2,2], k=1).odd_component() sage: c.rauzy_diagram() Rauzy diagram with 294 permutations sage: c.rauzy_class_cardinality() 294 sage: c = Stratum([2,2,0], k=1).odd_component() sage: c.rauzy_class_cardinality() 2723 sage: c.rauzy_diagram(left_degree=2) Rauzy diagram with 2352 permutations sage: c.rauzy_class_cardinality(left_degree=2) 2352 sage: c.rauzy_diagram(left_degree=2, reduced=False) Rauzy diagram with 7056 permutations sage: c.rauzy_class_cardinality(left_degree=2, reduced=False) 7056 sage: c.rauzy_diagram(left_degree=0) Rauzy diagram with 371 permutations sage: c.rauzy_class_cardinality(left_degree=0) 371 sage: c.rauzy_diagram(left_degree=0, reduced=False) Rauzy diagram with 6678 permutations sage: c.rauzy_class_cardinality(left_degree=0, reduced=False) 6678
Example in higher genus for which an explicit computation of the Rauzy diagram would be very long:
sage: c = Stratum([4,2,0], k=1).odd_component() sage: c.rauzy_class_cardinality() 262814 sage: c = Stratum([4,4,4], k=1).odd_component() sage: c.rauzy_class_cardinality() 24691288838 sage: c.rauzy_class_cardinality(left_degree=4, reduced=False) 1234564441900
- single_cylinder_representative(alphabet=None, reduced=True)#
Returns a single cylinder permutation representative.
Returns a permutation representative of a square-tiled surface in this component having a single vertical cylinder and a single horizontal cylinder.
Such representatives were constructed for every stratum of Abelian differentials by Jeffreys [Jef19].
INPUT:
alphabet– an optional alphabet for the permutation representativereduced(boolean, defaultTrue) – whether to return a reduced permutation (ie without labels)
EXAMPLES:
sage: from surface_dynamics import Stratum sage: cc = Stratum([4], k=1).odd_component() sage: p = cc.single_cylinder_representative(alphabet=Alphabet(name='upper')) sage: p A B C D E F C F E B D A sage: p.stratum_component() == cc True sage: cc = Stratum([6,2], k=1).odd_component() sage: p = cc.single_cylinder_representative() sage: p 0 1 2 3 4 5 6 7 8 9 10 2 5 4 6 3 8 10 7 1 9 0 sage: p.stratum_component() == cc True
- spin()#
Returns 1 which is, by definition, the spin parity of this stratum component.
EXAMPLES:
sage: from surface_dynamics import Stratum sage: c = Stratum([4], k=1).odd_component(); c H_3(4)^odd sage: c.spin() 1
- standard_permutations_number()#
Return the number of standard permutation of this even component.
EXAMPLES:
sage: from surface_dynamics import Stratum
In genus 2, there are two strata which contains an odd component:
sage: C = Stratum([4], k=1).odd_component() sage: len(C.standard_permutations()) 7 sage: C.standard_permutations_number() 7 sage: C = Stratum([2,2], k=1).odd_component() sage: len(C.standard_permutations()) 11 sage: C.standard_permutations_number() 11
In genus 3, the number of standard permutations is reasonably small and the whole set can be computed:
sage: C = Stratum([6], k=1).odd_component() sage: len(C.standard_permutations()) # long time 135 sage: C.standard_permutations_number() 135 sage: C = Stratum([4,2], k=1).odd_component() sage: len(C.standard_permutations()) # long time 472 sage: C.standard_permutations_number() 472 sage: C = Stratum([2,2,2], k=1).odd_component() sage: len(C.standard_permutations()) # long time 372 sage: C.standard_permutations_number() 372
For higher genera, this number can be very big:
sage: C = Stratum([8,6,4,2], k=1).odd_component() sage: C.standard_permutations_number() 26596699869748377600