strata

Strata of differential on Riemann surfaces

This file gather common code used in abelian_strata and quadratic_strata.

class surface_dynamics.flat_surfaces.strata.Strata

Strata of Abelian or Quadratic differentials.

class surface_dynamics.flat_surfaces.strata.Stratum(signature, k=1)

Stratum of holomorphic or meromorphic k-differentials on smooth connected Riemann surfaces.

INPUT:

• signature – a list of n integers (determine the angles of conical singularities)

• k – optional integer (default 1) the order of the differential

EXAMPLES:

sage: from surface_dynamics import Stratum
sage: Stratum((2,), k=1)
H_2(2)

TESTS:

sage: S = Stratum((2,), k=1)
sage: loads(dumps(S)) == S
True
sage: S = Stratum((2, 2, 2), k=3)
sage: loads(dumps(S)) == S
True

Test unique representation:

sage: assert Stratum([1, 2, 1], k=1) is Stratum((2, 1, 1), k=1)
sage: assert Stratum([1, 2, 1], k=2) is Stratum((2, 1, 1), k=2)
sage: assert Stratum([2, 4, 2], k=4) is Stratum((4, 2, 2), k=4)

components()

Lists the connected components of the Stratum.

OUTPUT:

list – a list of connected components of stratum

EXAMPLES:

sage: from surface_dynamics import Stratum

Some abelian strata:

sage: Stratum([0], k=1).components()
(H_1(0)^hyp,)
sage: Stratum([2], k=1).components()
(H_2(2)^hyp,)
sage: Stratum([4], k=1).components()
(H_3(4)^hyp, H_3(4)^odd)
sage: Stratum([2,2], k=1).components()
(H_3(2^2)^hyp, H_3(2^2)^odd)
sage: Stratum([1,1,1,1], k=1).components()
(H_3(1^4)^c,)

Some quadratic strata:

sage: Stratum([12], k=2).components()
(Q_4(12)^reg, Q_4(12)^irr)
sage: Stratum([6,-1,-1], k=2).components()
(Q_2(6, -1^2)^hyp, Q_2(6, -1^2)^nonhyp)

connected_components()

Return the connected components of this stratum of differentials.

• Abelian holomorphic differentials [KonZor03]

• Quadratic differentials with at most simple poles [Lan08]

dimension()

Return the complex dimension of this stratum.

The dimension is 2g-2+s+1 where g is the genus of surfaces in the stratum, s the number of singularities. The complex dimension of a stratum is also the number of intervals of any interval exchange transformations associated to the strata.

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum((0,), k=1).dimension()
2
sage: Stratum((0,0), k=1).dimension()
3
sage: Stratum((2,), k=1).dimension()
4
sage: Stratum((1,1), k=1).dimension()
5

sage: Stratum({-1:4}, k=2).dimension()
2

sage: Stratum((1, -1), k=1).dimension()
2

sage: Stratum((-1, -1, -2), k=2).dimension()
1
sage: Stratum((-2, -2, -2), k=3).dimension()
1

sage: a = Stratum((4,3,2,1,0), k=1)
sage: p = a.permutation_representative()
sage: len(p) == a.dimension()
True

genus()

is_connected()

Test if the strata is connected.

EXAMPLES:

sage: from surface_dynamics import Stratum

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

sage: Stratum([-1,-1,-1,-1], k=2).is_connected()
True
sage: Stratum([12], k=2).is_connected()
False

is_empty()

Return True if the stratum is empty

EXAMPLES:

sage: from surface_dynamics import Stratum

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

masur_veech_volume(rational=False, method=None)

Return the Masur-Veech volume of this stratum.

INPUT:

• rational (optional, boolean) - if False (default) return the Masur-Veech volume and if True return the Masur-Veech volume divided by zeta(2g).

• method (optional string) - the method to use to compute the volume either, see masur_veech_volume()

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum([2], k=1).masur_veech_volume()
1/120*pi^4
sage: Stratum([1,1,1,1], k=1).masur_veech_volume()
1/4860*pi^6
sage: Stratum([20], k=1).masur_veech_volume()
1604064377302075061983/792184445986404135075840000000000*pi^22

nb_fake_zeros()

Return the number of fake zeros.

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum([0], k=1).nb_fake_zeros()
doctest:warning
...
UserWarning: nb_fake_zeros() has been deprecated and will be removed in a future version of surface-dynamics; use signature()
1
sage: Stratum([1,1,0,0], k=1).nb_fake_zeros()
2

sage: Stratum([0,4,2,2], k=2).nb_fake_zeros()
1

nb_poles()

Return the number of poles of this quadratic stratum.

nb_zeros(fake_zeros=True, poles=True)

Returns the number of zeros of self.

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum([0], k=1).nb_zeros()
doctest:warning
...
UserWarning: nb_zero() has been deprecated and will be removed in a future version of surface-dynamics; use signature()
1
sage: Stratum({2:4,3:2}, k=1).nb_zeros()
6

sage: Stratum({-1:4}, k=2).nb_zeros()
4
sage: Stratum({-1:4,1:4}, k=2).nb_zeros()
8

number_of_components()

Returns the number of connected components of self

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum([2], k=1).number_of_components()
1
sage: Stratum([4], k=1).number_of_components()
2
sage: Stratum([3,3], k=1).number_of_components()
2

one_component()

Returns a connected component of this stratum.

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum([2], k=1).one_component()
H_2(2)^hyp

permutation_representative(*args, **kwds)

Return a permutation of interval exchanges associated to this stratum.

This method only makes sense for Abelian and quadratic differentials.

EXAMPLES:

sage: from surface_dynamics import Stratum

Examples from Abelian differentials:

sage: a = Stratum([3,2,1,0,0], k=1)
sage: p = a.permutation_representative()
sage: p.stratum()
H_4(3, 2, 1, 0^2)
sage: a = Stratum([2, 2, 2], k=1)
sage: p = a.permutation_representative()
sage: p.stratum()
H_4(2^3)

Examples from quadratic differentials:

sage: a = Stratum([6,-1,-1], k=2)
sage: p = a.permutation_representative()
sage: p.stratum()
Q_2(6, -1^2)
sage: a = Stratum([-1,-1,-1,-1,0,0], k=2)
sage: p = a.permutation_representative()
sage: p.stratum()
Q_0(0^2, -1^4)

random_component()

Returns a random connected component of this stratum.

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Q = Stratum([6,6], k=2)
sage: Q.random_component() # random
Q_4(6^2)^hyp
sage: Q.random_component() # random
Q_4(6^2)^reg

rank()

Return the rank of this GL(2,R)-invariant manifold (half dimension of the absolute part of the tangent space).

EXAMPLES:

sage: from surface_dynamics import Stratum, QuadraticStrata

sage: Stratum((0,0), k=1).rank()
1
sage: Stratum((2,), k=1).rank()
2
sage: Stratum((2,0,0), k=1).rank()
2

sage: Stratum({-1: 4}, k=2).rank()
1
sage: Stratum({-1:4, 0:5}, k=2).rank()
1

Complete list of rank 2 quadratic strata listed by dimension:

sage: for dim in range(4, 9):
....:     quad = [Q for Q in QuadraticStrata(dimension=dim) if Q.rank() == 2]
....:     print("%d: %s" % (dim, ", ".join(map(str, quad))))
4: Q_2(5, -1), Q_1(1^2, -1^2), Q_1(3, -1^3), Q_0(1, -1^5)
5: 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)
6: Q_3(6, 2), Q_3(4^2), Q_2(2^2, 1, -1), Q_2(4, 2, -1^2), Q_1(2^2, -1^4)
7: Q_3(4, 2^2), Q_2(2^3, -1^2)
8: Q_3(2^4)

signature()

Return the order of zeros with multiplicities.

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum([1, 2, 3], k=1).signature()
(3, 2, 1)
sage: Stratum({2: 4}, k=1).signature()
(2, 2, 2, 2)
sage: Stratum([-1, 1], k=1).signature()
(1, -1)

sage: Stratum({-1: 4}, k=2).signature()
(-1, -1, -1, -1)
sage: Stratum({1: 8}, k=2).signature()
(1, 1, 1, 1, 1, 1, 1, 1)
sage: Stratum({-2: 2, 0: 1}, k=2).signature()
(0, -2, -2)

surface_differential_order()

Return the order of differentials in this stratum.

EXAMPLES:

sage: from surface_dynamics import Stratum
sage: Stratum([1]*6, 3).surface_differential_order()
3

surface_genus()

Return the genus of the surfaces in this stratum.

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum((0,), k=1).surface_genus()
1
sage: Stratum((1,1), k=1).surface_genus()
2
sage: Stratum((3,2,1), k=1).surface_genus()
4

sage: Stratum((-1,-1,-1,-1), k=2).surface_genus()
0

surface_has_finite_area()

Return whether the k-differentials in this moduli space have finite or infinite area.

EXAMPLES:

sage: from surface_dynamics import Stratum
sage: Stratum((3, 2, 1), k=1).surface_has_finite_area()
True
sage: Stratum((1, 0, -1), k=1).surface_has_finite_area()
False

sage: Stratum([-1]*6, k=3).surface_has_finite_area()
True
sage: Stratum([-2]*3, k=3).surface_has_finite_area()
True
sage: Stratum([-3]*2, k=3).surface_has_finite_area()
False

unique_component()

Returns the unique component of self or raise a ValueError.

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: a = Stratum([1,1], k=1)
sage: a
H_2(1^2)
sage: a.unique_component()
H_2(1^2)^hyp

sage: a = Stratum([3,2,1], k=1)
sage: a
H_4(3, 2, 1)
sage: a.unique_component()
H_4(3, 2, 1)^c

sage: Stratum({1:1, -1:5}, k=2).unique_component()
Q_0(1, -1^5)^c
sage: Stratum([3,2,-1], k=2).unique_component()
Q_2(3, 2, -1)^nonhyp

sage: Stratum([12], k=2).unique_component()
Traceback (most recent call last):
...
ValueError: several components for this stratum

zeros(fake_zeros=True, poles=True)

class surface_dynamics.flat_surfaces.strata.StratumComponent(stratum)

Generic class for connected component of a stratum of flat surfaces.

Assumes there are implemented

• a method .permutation_representative()

There may be

• an attribute ._name

• an attribute ._latex_name

dimension()

Return the (complex) dimension of this GL(2,R)-invariant orbifold.

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum([4], k=1).odd_component().dimension()
6
sage: Stratum([12], k=2).regular_component().dimension()
7

genus()

masur_veech_volume(rational=False, method=None)

Return the Masur-Veech volume of this stratum component.

INPUT:

• rational (optional, boolean) - if False (default) return the Masur-Veech volume and if True return the Masur-Veech volume divided by zeta(2g).

• method (optional string) - the method to use to compute the volume either, see masur_veech_volume()

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum([4], k=1).hyperelliptic_component().masur_veech_volume()
1/6720*pi^6
sage: Stratum([6], k=1).even_component().masur_veech_volume()
32/1913625*pi^8

rank()

Return the rank of this GL(2,R)-invariant orbifold.

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: Stratum([4], k=1).odd_component().rank()
3
sage: Stratum([12], k=2).regular_component().rank()
3

stratum()

Return the stratum associated to self

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: a = Stratum([4,4], k=1)
sage: all([c.stratum() == a for c in a.components()])
True

surface_differential_order()

surface_genus()

Return genus of the corresponding stratum

EXAMPLES:

sage: from surface_dynamics import Stratum

sage: a = Stratum([4,4], k=1)
sage: a.one_component().surface_genus()
5

surface_has_finite_area()

surface_dynamics.flat_surfaces.strata.list_to_exp_list(l)

Convert list into exponential notation.

EXAMPLES:

sage: from surface_dynamics import *

sage: from surface_dynamics.flat_surfaces.strata import list_to_exp_list
sage: l = [0,0,2,2,3,2,0,0,0]
sage: list_to_exp_list(l)
[(0, 2), (2, 2), (3, 1), (2, 1), (0, 3)]