quadratic_strata

Strata of quadratic differentials on Riemann surfaces

More precisely, we are interested in meromorphic quadratic differentials with at most simple poles on closed compact connected Riemann surfaces, which are not globally the square of an abelian differential.

The moduli space of such quadratic differentials on Riemann surfaces of a given genus is a complex orbifold, stratified by the degrees of zeros (zeros of degree -1, or simple poles, being allowed). The strata themselves are complex orbifolds. Most strata are connected but some (infinitely many) are not.

A stratum corresponds to the Sage object QuadraticStratum.

The classification of connected components of strata of quadratic differentials was established by Erwan Lanneau in [Lan08], after a similar classification was established by Kontsevich and Zorich in [KonZor03] in the Abelian case.

Each stratum has one or two connected components and each component is associated to an extended Rauzy class. The components() method gives the decomposition of a stratum into its connected components.

A representative for each connected component of stratum is given by Zorich in [Zor08].

This is implemented here following [Zor08]:

• genus zero stratum permutation_representative()

• genus one stratum permutation_representative()

• genus two hyperellitic component permutation_representative()

• genus two non-hyperellitic component permutation_representative()

• connected component permutation_representative()

• hyperelliptic component permutation_representative()

• non-hyperelliptic component is similar to connected components

• regular component of exceptional stratum permutation_representative()

• irregular component of exceptional stratum permutation_representative()

The inverse operation, i.e., starting from a permutation, determine the connected component it lives in, is partially written in [KonZor03]. See: stratum_component().

The code here implements the descriptions in [Zor08]. Zorich already implemented all this for Mathematica in [ZS].

See also abelian_strata for Abelian strata.

AUTHORS:

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

• Samuel Lelievre (2010-10-08): quadratic strata

EXAMPLES:

sage: from surface_dynamics import *

Construction of a stratum from a list of singularity degrees:

sage: a = Stratum([2,2], k=2)
sage: a
Q_2(2^2)
sage: a.surface_genus()
2

sage: a = Stratum([4,3,2,2,1], k=2)
sage: a
Q_4(4, 3, 2^2, 1)
sage: a.surface_genus()
4

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

sage: a1 = Stratum([7,5,3,1], k=2)
sage: a1
Q_5(7, 5, 3, 1)
sage: a2 = Stratum([3,1,7,5], k=2)
sage: a2
Q_5(7, 5, 3, 1)
sage: a1 == a2
True

List the connected components of a stratum:

sage: a = Stratum([6,2], k=2)
sage: a.components()
(Q_3(6, 2)^hyp, Q_3(6, 2)^nonhyp)

sage: a = Stratum([12], k=2)
sage: cc = a.components()
sage: cc
(Q_4(12)^reg, Q_4(12)^irr)
sage: for c in cc:
....:     print(c)
....:     print(c.permutation_representative())
Q_4(12)^reg
0 1 2 1 2 3 4 3 4 5
5 6 7 6 7 0
Q_4(12)^irr
0 1 2 3 4 5 6 5
7 6 4 7 3 2 1 0

sage: a = Stratum([1, 1, 1, 1], k=2)
sage: a.components()
(Q_2(1^4)^hyp,)
sage: c = a.components()[0]
sage: p = c.permutation_representative(); p
0 1 2 3 1 4 5
2 6 5 4 6 3 0

surface_dynamics.flat_surfaces.quadratic_strata.CQSC

alias of ConnectedQuadraticStratumComponent

class surface_dynamics.flat_surfaces.quadratic_strata.ConnectedQuadraticStratumComponent(stratum)

Connected component of stratum of quadratic differentials.

This class is intended to be called internally rather than directly. Call only with appropriate parameters, in particular correct genus: no consistency check inside, no prediction as to what may happen otherwise.

permutation_representative(reduced=True, alphabet=None, relabel=True)

Returns a generalized permutation representative.

NOTES:

The representative is made by constructing two lists l0 and l1 which correspond to a generalized permutation representative for the stratum with simple zeros, and then erasing some elements from l0 and l1 (this corresponds to collapsing saddle connections to merge zeros). It may be possible to find a faster way to obtain the desired l0 and l1 than by constructing the long versions and erasing symbols; the current implementation has a loop with “del l0[l0.index(i)]” and “del l1[l1.index(i)]”.

EXAMPLES:

sage: from surface_dynamics import *

sage: cc = Stratum([6,-1,-1], k=2).non_hyperelliptic_component()
sage: p = cc.permutation_representative(); p
0 1 2 1 3
3 4 4 5 5 2 0
sage: p.stratum_component()
Q_2(6, -1^2)^nonhyp

sage: cc = Stratum({3: 2, -1: 2}, k=2).non_hyperelliptic_component()
sage: p = cc.permutation_representative(); p
0 1 2 1 3
4 3 5 5 4 6 6 2 0
sage: p.stratum_component()
Q_2(3^2, -1^2)^nonhyp

sage: cc = Stratum([8], k=2).unique_component()
sage: p = cc.permutation_representative(); p
0 1 2 1 2 3
4 5 4 5 3 0
sage: p.stratum_component()
Q_3(8)^c

sage: Q = Stratum([4,4], k=2).unique_component()
sage: p = Q.permutation_representative(); p
0 1 2 3 2 3 4
5 6 5 6 1 4 0
sage: p.stratum_component()
Q_3(4^2)^c

sage: Q = Stratum({12:1,-1:4}, k=2).unique_component()
sage: p = Q.permutation_representative()
sage: p
0 1 2 1 2 3 3 4 4 5 5 6 6 7
8 9 8 9 7 0
sage: p.stratum()
Q_3(12, -1^4)

surface_dynamics.flat_surfaces.quadratic_strata.DeprecatedQuadraticStratumConstructor(*l, **kwds)

TESTS:

sage: from surface_dynamics import *
sage: QuadraticStratum(-1,-1,-1,-1) is QuadraticStratum({-1:4})
doctest:warning
...
UserWarning: QuadraticStratum has changed its arguments in order to handle meromorphic and higher-order differentials; use Stratum instead
True
sage: QuadraticStratum(-1,-1,-1,-1) is Stratum((-1,-1,-1,-1), 2)
True

surface_dynamics.flat_surfaces.quadratic_strata.GOQSC

alias of GenusOneQuadraticStratumComponent

surface_dynamics.flat_surfaces.quadratic_strata.GTHQSC

alias of GenusTwoHyperellipticQuadraticStratumComponent

surface_dynamics.flat_surfaces.quadratic_strata.GTNQSC

alias of GenusTwoNonhyperellipticQuadraticStratumComponent

surface_dynamics.flat_surfaces.quadratic_strata.GZQSC

alias of GenusZeroQuadraticStratumComponent

class surface_dynamics.flat_surfaces.quadratic_strata.GenusOneQuadraticStratumComponent(stratum)

This class is intended to be called internally rather than directly. Call only with appropriate parameters, in particular correct genus: no consistency check inside, no prediction as to what may happen otherwise.

permutation_representative(reduced=True, alphabet=None, relabel=True)

Returns a generalized permutation representative.

EXAMPLES:

sage: from surface_dynamics import *

sage: Stratum({2: 1, -1: 2}, k=2).permutation_representative()
0 1 2 2
1 3 3 0
sage: Stratum({3: 1, -1: 3}, k=2).permutation_representative()
0 1 2 2 3 3
1 4 4 0
sage: Stratum({1: 2, -1: 2}, k=2).permutation_representative()
0 1 2 3 3
2 1 4 4 0
sage: Q = Stratum({2: 1, 1: 1, -1: 3}, k=2)
sage: Q.permutation_representative(alphabet='abcdef')
a b c c d e e
d b f f a

TESTS:

sage: from surface_dynamics import *

sage: Stratum({1: 1, -1: 1}, k=2).permutation_representative()
Traceback (most recent call last):
...
EmptySetError: The stratum is empty

sage: Q = Stratum([2,-1,-1,0], k=2)
sage: p = Q.permutation_representative()
sage: p
0 1 2 3 3
2 4 4 0 1
sage: p.stratum() == Q
True

class surface_dynamics.flat_surfaces.quadratic_strata.GenusTwoHyperellipticQuadraticStratumComponent(stratum)

This class is intended to be called internally rather than directly. Call only with appropriate parameters, in particular correct genus: no consistency check inside, no prediction as to what may happen otherwise.

permutation_representative(reduced=True, alphabet=None, relabel=True)

Returns a generalized permutation representative.

TESTS:

sage: from surface_dynamics import *

sage: Q = Stratum({1: 4}, k=2) sage: H = Q.hyperelliptic_component() sage: H.permutation_representative() 0 1 2 3 1 4 5 2 6 5 4 6 3 0

sage: Q = Stratum([2,1,1], k=2) sage: H = Q.hyperelliptic_component() sage: H.permutation_representative(alphabet=’abcdef’) a b c b d e f e d f c a

sage: Q = Stratum([2,2], k=2) sage: H = Q.hyperelliptic_component() sage: H.permutation_representative() 0 1 2 1 3 4 3 4 2 0

class surface_dynamics.flat_surfaces.quadratic_strata.GenusTwoNonhyperellipticQuadraticStratumComponent(stratum)

This class is intended to be called internally rather than directly. Call only with appropriate parameters, in particular correct genus: no consistency check inside, no prediction as to what may happen otherwise.

permutation_representative(reduced=True, alphabet=None, relabel=True)

Returns a generalized permutation representative.

TESTS:

sage: from surface_dynamics import *

sage: Q = Stratum({6: 1, -1: 2}, k=2)
sage: N = Q.non_hyperelliptic_component()
sage: N.permutation_representative()
0 1 2 1 3
3 4 4 5 5 2 0
sage: Q = Stratum({3: 2, -1: 2}, k=2)
sage: N = Q.non_hyperelliptic_component()
sage: N.permutation_representative()
0 1 2 1 3
4 3 5 5 4 6 6 2 0

class surface_dynamics.flat_surfaces.quadratic_strata.GenusZeroQuadraticStratumComponent(stratum)

This class is intended to be called internally rather than directly. Call only with appropriate parameters, in particular correct genus: no consistency check inside, no prediction as to what may happen otherwise.

permutation_representative(reduced=True, alphabet=None, relabel=True)

Returns a generalized permutation representative.

EXAMPLES:

sage: from surface_dynamics import *

sage: Stratum([-1,-1,-1,-1], k=2).permutation_representative()
0 1 1
2 2 0
sage: Stratum({1:1, -1:5}, k=2).permutation_representative()
0 1 1
2 2 3 3 4 4 0
sage: Stratum({2:1, -1:6}, k=2).permutation_representative()
0 1 1
2 2 3 3 4 4 5 5 0
sage: Stratum({1:2, -1:6}, k=2).permutation_representative()
0 1 1
2 2 3 4 4 5 5 3 6 6 0
sage: Stratum({2:1, 1:1, -1:7}, k=2).permutation_representative()
0 1 1
2 2 3 3 4 5 5 6 6 4 7 7 0

surface_dynamics.flat_surfaces.quadratic_strata.HQSC

alias of HyperellipticQuadraticStratumComponent

class surface_dynamics.flat_surfaces.quadratic_strata.HyperellipticQuadraticStratumComponent(stratum)

This class is intended to be called internally rather than directly. Call only with appropriate parameters, in particular correct genus: no consistency check inside, no prediction as to what may happen otherwise.

permutation_representative(reduced=True, alphabet=None, relabel=True)

Returns a generalized permutation representative.

EXAMPLES:

sage: from surface_dynamics import *

sage: cc = Stratum({6:1, -1:2}, k=2).hyperelliptic_component()
sage: cc.permutation_representative()
0 1 2 3 4 1
5 4 3 2 5 0
sage: cc = Stratum({3: 2, -1: 2}, k=2).hyperelliptic_component()
sage: cc.permutation_representative()
0 1 2 3 4 5 1
6 5 4 3 2 6 0
sage: cc = Stratum([10,10], k=2).hyperelliptic_component()
sage: cc.permutation_representative()
0 1 2 3 4 5 6 1 7 8 9 10 11
11 10 9 8 7 12 6 5 4 3 2 12 0

surface_dynamics.flat_surfaces.quadratic_strata.IEQSC

alias of IrregularExceptionalQuadraticStratumComponent

class surface_dynamics.flat_surfaces.quadratic_strata.IrregularExceptionalQuadraticStratumComponent(stratum)

This class is intended to be called internally rather than directly. Call only with appropriate parameters, in particular correct genus: no consistency check inside, no prediction as to what may happen otherwise.

permutation_representative(reduced=True, alphabet=None, relabel=True)

Returns a generalized permutation representative.

EXAMPLES:

sage: from surface_dynamics import *

sage: cc = Stratum([9,-1], k=2).irregular_component()
sage: p = cc.permutation_representative(); p
0 1 2 3 4 1 2 3 4 5
5 6 6 0
sage: p.stratum_component()  # optional
Q_3(9, -1)^irr

sage: cc = Stratum([6,3,-1], k=2).irregular_component()
sage: p = cc.permutation_representative(); p
0 1 2 3 4 5 1 2 3 4 5 6
6 7 7 0
sage: p.stratum_component()
Q_3(6, 3, -1)^irr

sage: cc = Stratum([12], k=2).irregular_component()
sage: p = cc.permutation_representative(); p
0 1 2 3 4 5 6 5
7 6 4 7 3 2 1 0
sage: p.stratum_component()
Q_4(12)^irr

surface_dynamics.flat_surfaces.quadratic_strata.NQSC

alias of NonhyperellipticQuadraticStratumComponent

class surface_dynamics.flat_surfaces.quadratic_strata.NonhyperellipticQuadraticStratumComponent(stratum)

Non hyperelliptic component of stratum of quadratic differentials.

surface_dynamics.flat_surfaces.quadratic_strata.QSC

alias of QuadraticStratumComponent

surface_dynamics.flat_surfaces.quadratic_strata.QuadraticStrata(genus=None, dimension=None, min_nb_poles=None, max_nb_poles=None, nb_poles=None, fake_zeros=False)

Quadratic strata.

INPUT:

• genus - a non negative integer or None

• dimension - a non negative integer or None

• min_nb_poles, max_nb_poles - the minimum and maximum number of poles allowed

• nb_poles - integer - the number of poles (if the option is set then the options min_nb_poles and max_nb_poles are ignored)

• fake_zeros - boolean - whether to allow fake zeros or not

EXAMPLES:

sage: from surface_dynamics import *

sage: Q = QuadraticStrata(genus=2); Q
Quadratic strata of genus 2 surfaces
sage: Q.cardinality()
+Infinity
sage: i = iter(Q)
sage: next(i)
Q_2(2^2)
sage: next(i)
Q_2(2, 1^2)
sage: next(i)
Q_2(1^4)
sage: next(i)
Q_2(5, -1)

sage: Q = QuadraticStrata(dimension=5); Q
Quadratic strata of dimension 5

sage: Q = QuadraticStrata(genus=3,max_nb_poles=6); Q
Quadratic strata of genus 3 surfaces with at most 6 poles
sage: Q.cardinality()
463
sage: for q in Q: print(q)
Q_3(8)
Q_3(7, 1)
Q_3(6, 2)
...
Q_3(2^2, 1^10, -1^6)
Q_3(2, 1^12, -1^6)
Q_3(1^14, -1^6)

sage: Q = QuadraticStrata(genus=2,nb_poles=0); Q
Quadratic strata of genus 2 surfaces with no pole
sage: for q in Q: print(q)
Q_2(2^2)
Q_2(2, 1^2)
Q_2(1^4)


sage: Q = QuadraticStrata(dimension=7,min_nb_poles=1,max_nb_poles=3); Q
Quadratic strata of dimension 7 with at least 1 and at most 3 poles
sage: for q in Q: print(q)
Q_3(8, 1, -1)
Q_3(7, 2, -1)
Q_3(6, 3, -1)
Q_3(5, 4, -1)
Q_2(2, 1^3, -1)
Q_3(10, -1^2)
Q_2(4, 1^2, -1^2)
Q_2(3, 2, 1, -1^2)
Q_2(2^3, -1^2)
Q_2(6, 1, -1^3)
Q_2(5, 2, -1^3)
Q_2(4, 3, -1^3)

sage: Q = QuadraticStrata(dimension=6, genus=0)
sage: Q
Quadratic strata of genus 0 surfaces and dimension 6
sage: for q in Q: print(q)
Q_0(1^2, -1^6)
Q_0(3, -1^7)

sage: Q = QuadraticStrata(dimension=5, genus=1, fake_zeros=True, nb_poles=0)
sage: for q in Q: print(q)

class surface_dynamics.flat_surfaces.quadratic_strata.QuadraticStrata_class

Base class for strata of quadratic differentials.

class surface_dynamics.flat_surfaces.quadratic_strata.QuadraticStrata_d(dimension, min_nb_poles, max_nb_poles, fake_zeros)

Strata with prescribed dimension.

EXAMPLES:

sage: from surface_dynamics import *

sage: for q in QuadraticStrata(dimension=5): print(q)
Q_3(8)
Q_2(2, 1^2)
Q_2(4, 1, -1)
Q_2(3, 2, -1)
Q_2(6, -1^2)
Q_1(2, 1, -1^3)
Q_1(4, -1^4)
Q_0(2, -1^6)

sage: Q = QuadraticStrata(dimension=6,nb_poles=1); Q
Quadratic strata of dimension 6 with 1 pole
sage: for q in Q: print(q)
Q_3(9, -1)
Q_2(3, 1^2, -1)
Q_2(2^2, 1, -1)

class surface_dynamics.flat_surfaces.quadratic_strata.QuadraticStrata_g(genus, min_nb_poles=None, max_nb_poles=None)

Stratas of genus g surfaces.

EXAMPLES:

sage: from surface_dynamics import *

sage: Q = QuadraticStrata(genus=3); Q
Quadratic strata of genus 3 surfaces
sage: Q.cardinality()
+Infinity
sage: i = iter(Q)
sage: next(i)
Q_3(8)

sage: Q = QuadraticStrata(genus=2,max_nb_poles=1); Q
Quadratic strata of genus 2 surfaces with at most 1 pole
sage: Q.list()
[Q_2(2^2), Q_2(2, 1^2), Q_2(1^4), Q_2(5, -1), Q_2(4, 1, -1), Q_2(3, 2, -1), Q_2(3, 1^2, -1), Q_2(2^2, 1, -1), Q_2(2, 1^3, -1), Q_2(1^5, -1)]

sage: Q = QuadraticStrata(genus=4,nb_poles=3); Q
Quadratic strata of genus 4 surfaces with 3 poles
sage: Q.cardinality()
176

an_element_()

Return the first element of this list of strata.

EXAMPLES:

sage: from surface_dynamics import *

sage: Q = QuadraticStrata(genus=4); Q
Quadratic strata of genus 4 surfaces
sage: Q.first()
Q_4(12)

sage: Q = QuadraticStrata(genus=3,nb_poles=1); Q
Quadratic strata of genus 3 surfaces with 1 pole
sage: Q.first()
Q_3(9, -1)

sage: Q = QuadraticStrata(genus=3,min_nb_poles=2); Q
Quadratic strata of genus 3 surfaces with at least 2 poles
sage: Q.first()
Q_3(10, -1^2)

first()

Return the first element of this list of strata.

EXAMPLES:

sage: from surface_dynamics import *

sage: Q = QuadraticStrata(genus=4); Q
Quadratic strata of genus 4 surfaces
sage: Q.first()
Q_4(12)

sage: Q = QuadraticStrata(genus=3,nb_poles=1); Q
Quadratic strata of genus 3 surfaces with 1 pole
sage: Q.first()
Q_3(9, -1)

sage: Q = QuadraticStrata(genus=3,min_nb_poles=2); Q
Quadratic strata of genus 3 surfaces with at least 2 poles
sage: Q.first()
Q_3(10, -1^2)

last()

Return the last element of this list of strata.

EXAMPLES:

sage: from surface_dynamics import *

sage: Q = QuadraticStrata(genus=2, nb_poles=0); Q
Quadratic strata of genus 2 surfaces with no pole
sage: Q.last()
Q_2(1^4)

sage: Q = QuadraticStrata(genus=0); Q
Quadratic strata of genus 0 surfaces
sage: Q.last()
Traceback (most recent call last):
...
NotImplementedError: infinite list

TESTS:

sage: from surface_dynamics import *

sage: Q = QuadraticStrata(genus=1,nb_poles=2)
sage: Q.list()[-1] == Q.last()
True

sage: Q = QuadraticStrata(genus=1,max_nb_poles=2)
sage: Q.list()[-1] == Q.last()
True

sage: Q = QuadraticStrata(genus=0,nb_poles=10)
sage: Q.list()[-1] == Q.last()
True

class surface_dynamics.flat_surfaces.quadratic_strata.QuadraticStrata_gd(genus, dimension, min_nb_poles, max_nb_poles, fake_zeros)

Quadratic strata with presrcribed genus and dimension.

class surface_dynamics.flat_surfaces.quadratic_strata.QuadraticStratum(signature, k=1)

Stratum of quadratic differentials.

EXAMPLES:

sage: from surface_dynamics import *

sage: Q = Stratum([15,-1,-1,-1], k=2)
sage: Q
Q_4(15, -1^3)
sage: Q.components()
(Q_4(15, -1^3)^c,)

sage: Q = Stratum([6,6], k=2)
sage: Q
Q_4(6^2)
sage: Q.components()
(Q_4(6^2)^hyp, Q_4(6^2)^reg, Q_4(6^2)^irr)

has_hyperelliptic_component()

Returns True if and only if self has a connected component which contains only hyperelliptic surfaces.

EXAMPLES:

sage: from surface_dynamics import *

sage: Stratum([2,2], k=2).has_hyperelliptic_component()
True
sage: Stratum([3,1], k=2).has_hyperelliptic_component()
False

has_non_hyperelliptic_component()

Test whether this stratum has a non hyperelliptic component.

EXAMPLES:

sage: from surface_dynamics import *

sage: Stratum([10,10], k=2).has_non_hyperelliptic_component()
True
sage: Stratum([6,6], k=2).has_non_hyperelliptic_component()
False

has_regular_and_irregular_components()

Test whether this component has a pair of regular, irregular components.

The list of strata of quadratic differentials that admit such a pair are:

in genus 3: Q(9,-1), Q(6,3,-1), Q(3,3,3,-1) in genus 4: Q(12), Q(9,3), Q(6,6), Q(6,3,3), Q(3,3,3,3)

EXAMPLES:

sage: from surface_dynamics import *

 sage: Stratum([9,-1], k=2).has_regular_and_irregular_components()
 True
 sage: Stratum([11,1], k=2).has_regular_and_irregular_components()
 False

hyperelliptic_component()

Returns the hyperelliptic component of self (or raise a ValueError).

EXAMPLES:

sage: from surface_dynamics import *

sage: Stratum([2,2], k=2).hyperelliptic_component()
Q_2(2^2)^hyp

sage: Stratum([3,1], k=2).hyperelliptic_component()
Traceback (most recent call last):
...
ValueError: the stratum has no hyperelliptic component

irregular_component()

Returns the irregular component of that stratum or raise a ValueError.

EXAMPLES:

sage: from surface_dynamics import *

 sage: Stratum([3,3,3,-1], k=2).irregular_component()
 Q_3(3^3, -1)^irr
 sage: Stratum([2,2], k=2).irregular_component()
 Traceback (most recent call last):
 ...
 ValueError: no irregular component for this stratum

non_hyperelliptic_component()

Returns the non hyperelliptic component of this stratum (or raise a ValueError).

EXAMPLES:

sage: from surface_dynamics import *

sage: Stratum([10,10], k=2).non_hyperelliptic_component()
Q_6(10^2)^nonhyp
sage: Stratum([2,2], k=2).non_hyperelliptic_component()
Traceback (most recent call last):
...
ValueError: no non hyperelliptic component

orientation_cover(fake_zeros=False)

Return the stratum of Abelian differentials which contains the set of orientation cover of quadratic differentials in this stratum.

OPTIONS:

• fake_zeros - boolean - if True, add fake zeros which corresponds to the double cover of poles.

EXAMPLES:

sage: from surface_dynamics import *

sage: q = Stratum({4:1,-1:4}, k=2)
sage: q
Q_1(4, -1^4)
sage: a1 = q.orientation_cover(); a1
H_3(2^2)
sage: q.orientation_cover(fake_zeros=True)
H_3(2^2, 0^4)

For hyperelliptic strata (orientation cover of quadratic strata of the form Q(n,-1^{n+4})) the dimension coincide. From [Lan08] we know that it only happens for those ones:

sage: q = Stratum({4:1, -1:8}, k=2)
sage: q
Q_0(4, -1^8)
sage: q.dimension()
7
sage: q.orientation_cover().dimension()
7

sage: q = Stratum({3:1, 1:1, -1:8}, k=2)
sage: q
Q_0(3, 1, -1^8)
sage: q.dimension()
8
sage: a = q.orientation_cover(); a
H_4(4, 2)
sage: a.dimension()
9

TESTS:

sage: Stratum({-1:4}, k=2).orientation_cover()
H_1(0)

random_cylindric_permutation()

Return a random cylindric permutation that belongs to this stratum.

EXAMPLES:

sage: from surface_dynamics import *

 sage: Q = Stratum([4,4], k=2)
 sage: Q.random_cylindric_permutation()  # random
 0 1 2 3 4 5 1 5
 2 6 3 6 4 0

regular_component()

Returns the regular component of that stratum or raise a ValueError.

EXAMPLES:

sage: from surface_dynamics import *

 sage: Stratum([12], k=2).regular_component()
 Q_4(12)^reg
 sage: Stratum([2,2,2,2], k=2).regular_component()
 Traceback (most recent call last):
 ...
 ValueError: no regular component for this stratum

spin()

Return the spin structure (None, 0 or 1) of that component.

Any quadratic differential has a canonic double cover (called the orientation cover) which is a surface, generally of higher genera, with an Abelian differential. The spin structure of the quadratic differential is the spin structure of that double cover.

The spin is None if any degree of zero of the quadratic differential is congruent to 2 mod 4. Otherwise, denoting respectively k1 and k3 the number of degree of zero congruent to 1 and 3 modulo 4 we have

..MATH:

spin = ((k1-k3)/4) mod 2

The proof of that formula is the object of [Lan04].

EXAMPLES:

sage: from surface_dynamics import *

sage: Stratum([1,3], k=2).spin()
0
sage: Stratum({1:1,3:1,-1:4}, k=2).spin()
1
sage: Stratum([2,2], k=2).spin() is None
True

class surface_dynamics.flat_surfaces.quadratic_strata.QuadraticStratumComponent(stratum)

Generic class for component of quadratic stratum.

lyapunov_exponents_H_minus(**kargs)

Compute the H^- Lyapunov exponents.

EXAMPLES:

sage: from surface_dynamics import *

sage: Q = Stratum({1:3, -1:3}, k=2).unique_component()
sage: Q.lyapunov_exponents_H_minus(nb_iterations=2**21) # long time # abs tol .05
[1.000, 0.369, 0.176]

sage: R = Stratum([3,3,3,-1], k=2).regular_component()
sage: R.lyapunov_exponents_H_minus(nb_iterations=2**21) # long time # abs tol .05
[1.000, 0.328, 0.1899, 0.0820]

lyapunov_exponents_H_plus(*args, **kargs)

Compute the H^+ part of Lyapunov exponents spectrum.

All arguments and keywords are sent to ?

EXAMPLES:

sage: from surface_dynamics import *

sage: R = Stratum([3,3,3,-1], k=2).regular_component()
sage: R.lyapunov_exponents_H_plus(nb_iterations=2**21) # long time # abs tol .05
[0.596, 0.405, 0.202]
sage: sum(_) # long time # abs tol .05
1.2

sage: R = Stratum([2,2,2,2], k=2).unique_component()
sage: R.lyapunov_exponents_H_plus(nb_iterations=2**21) # long time # abs tol .05
[0.651, 0.469, 0.243]
sage: sum(_) # long time # abs tol .05
1.3636

one_cylinder_diagram()

Return a separatrix diagram with one cylinder that belongs to this component of stratum.

EXAMPLES:

sage: from surface_dynamics import *

sage: Q = Stratum({1:1,-1:5}, k=2)
sage: c = Q.unique_component().one_cylinder_diagram()
sage: c
(0,0,1,1,2,2)-(3,3)
sage: c.stratum() == Q
True

sage: Q = Stratum([5,-1], k=2)
sage: c = Q.unique_component().one_cylinder_diagram()
sage: c
(0,1,1,2)-(3,0,3,2)
sage: c.stratum() == Q
True

sage: Stratum({-1:4}, k=2).unique_component().one_cylinder_diagram()
(0,0)-(1,1)
sage: Stratum({-1:4,0:1}, k=2).unique_component().one_cylinder_diagram()
(0,0)-(1,1,2,2)

orientation_cover_component(fake_zeros=False)

Return the connected component of Abelian stratum component which contains the set of orientation cover of quadratic differentials in this connected component.

ALGORITHM:

The spin only depends on the component and the only components for which the double cover belongs to a hyperelliptic component are Q(k,-1^k+4) and Q(2g-1,2g-1,-1,-1)^hyp

OPTIONS:

• fake_zeros - boolean - if True, add fake zeros which corresponds to the double cover of poles.

EXAMPLES:

sage: from surface_dynamics import *

 sage: cc = Stratum({5:1, -1:9}, k=2).unique_component()
 sage: cc.orientation_cover_component()
 H_4(6)^hyp

 sage: cc = Stratum({5:1, -1:5}, k=2).unique_component()
 sage: cc.orientation_cover_component()
 H_4(6)^odd

 sage: cc = Stratum({5:1, -1:1}, k=2).unique_component()
 sage: cc.orientation_cover_component()
 H_4(6)^even

 sage: cc = Stratum({1: 2, -1: 6}, k=2).unique_component()
 sage: cc.orientation_cover_component()
 H_3(2^2)^odd

 sage: cc = Stratum({4:1, -1:8}, k=2).unique_component()
 sage: cc.orientation_cover_component()
 H_3(2^2)^hyp

 sage: cc = Stratum({1:2, -1:2}, k=2).unique_component()
 sage: cc.orientation_cover_component()
 H_3(2^2)^hyp

random_cylindric_permutation(nsteps=64)

Return a cylindric permutation of the form p = ((0,...),(..., 0)) where 0 can be any label.

EXAMPLES:

sage: from surface_dynamics import *

 sage: Q = Stratum({4:1,-1:4}, k=2)
 sage: Q
 Q_1(4, -1^4)
 sage: c = Q.unique_component()
 sage: p = c.random_cylindric_permutation()
 sage: p.stratum_component()
 Q_1(4, -1^4)^c

 sage: Q = Stratum([6,6], k=2)
 sage: c_hyp, c_reg, c_irr = Q.components()
 sage: (c_hyp, c_reg, c_irr)
 (Q_4(6^2)^hyp, Q_4(6^2)^reg, Q_4(6^2)^irr)

 sage: all(c_hyp.random_cylindric_permutation().stratum_component() == c_hyp for _ in range(4))
 True
 sage: all(c_reg.random_cylindric_permutation().stratum_component() == c_reg for _ in range(4))
 True
 sage: all(c_irr.random_cylindric_permutation().stratum_component() == c_irr for _ in range(4))
 True

surface_dynamics.flat_surfaces.quadratic_strata.REQSC

alias of RegularExceptionalQuadraticStratumComponent

class surface_dynamics.flat_surfaces.quadratic_strata.RegularExceptionalQuadraticStratumComponent(stratum)

This class is intended to be called internally rather than directly. Call only with appropriate parameters, in particular correct genus: no consistency check inside, no prediction as to what may happen otherwise.

permutation_representative(reduced=True, alphabet=None, relabel=True)

Returns a generalized permutation representative.

EXAMPLES:

sage: from surface_dynamics import *

sage: cc = Stratum([9,-1], k=2).regular_component()
sage: p = cc.permutation_representative(); p
0 1 2 1 2 3 3 4
5 6 5 6 4 0
sage: p.stratum_component()
Q_3(9, -1)^reg

sage: cc = Stratum([6,3,-1], k=2).regular_component()
sage: p = cc.permutation_representative(); p
0 1 2 3 1 2 4 4 5
6 7 6 7 3 5 0
sage: p.stratum_component()
Q_3(6, 3, -1)^reg

sage: cc = Stratum([12], k=2).regular_component()
sage: p = cc.permutation_representative(); p
0 1 2 1 2 3 4 3 4 5
5 6 7 6 7 0
sage: p.stratum_component()
Q_4(12)^reg