Dynamical semigroups

A dynamical semigroup is a finitely generated subsemigroup of the endomorphism ring of a subscheme of projective or affine space.

AUTHORS:

• Dang Phan (August 6th, 2023): initial implementation

class sage.dynamics.arithmetic_dynamics.dynamical_semigroup.DynamicalSemigroup(systems)

Bases: Parent

A dynamical semigroup defined by a multiple dynamical systems on projective or affine space.

INPUT:

• ds_data – list or tuple of dynamical systems or objects that define dynamical systems

OUTPUT:

DynamicalSemigroup_affine if ds_data only contains dynamical systems over affine space; and DynamicalSemigroup_projective otherwise.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: DynamicalSemigroup(([x, y], [x^2, y^2]))
Dynamical semigroup over Projective Space of dimension 1 over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x : y)
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2)

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSystem([x, y], P)
sage: g = DynamicalSystem([x^2, y^2], P)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Projective Space of dimension 1 over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x : y)
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2)

sage: A.<x> = AffineSpace(QQ, 1)
sage: f = DynamicalSystem_affine(x, A)
sage: DynamicalSemigroup(f)
Dynamical semigroup over Affine Space of dimension 1 over Rational Field
 defined by 1 dynamical system:
  Dynamical System of Affine Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x) to (x)

sage: A.<x> = AffineSpace(QQ, 1)
sage: f = DynamicalSystem(x, A)
sage: g = DynamicalSystem(x^2, A)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Affine Space of dimension 1 over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Affine Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x) to (x)
  Dynamical System of Affine Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x) to (x^2)

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: X = P.subscheme(x - y)
sage: f = DynamicalSystem_projective([x, y], X)
sage: g = DynamicalSystem_projective([x^2, y^2], X)
sage: DynamicalSemigroup_projective([f, g])
Dynamical semigroup over Closed subscheme of Projective Space of dimension 1
 over Rational Field defined by: x - y
 defined by 2 dynamical systems:
  Dynamical System of Closed subscheme of Projective Space of dimension 1
   over Rational Field defined by: x - y
     Defn: Defined on coordinates by sending (x : y) to (x : y)
  Dynamical System of Closed subscheme of Projective Space of dimension 1
   over Rational Field defined by: x - y
     Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2)

If a dynamical semigroup is built from dynamical systems with different base rings, all systems will be coerced to the largest base ring:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: Q.<z,w> = ProjectiveSpace(RR, 1)
sage: f = DynamicalSystem([x, y], P)
sage: g = DynamicalSystem([z^2, w^2], Q)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Projective Space of dimension 1 over
 Real Field with 53 bits of precision defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1
   over Real Field with 53 bits of precision
    Defn: Defined on coordinates by sending (x : y) to (x : y)
  Dynamical System of Projective Space of dimension 1
   over Real Field with 53 bits of precision
    Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2)

sage: A.<x> = AffineSpace(QQ, 1)
sage: B.<y> = AffineSpace(RR, 1)
sage: f = DynamicalSystem(x, A)
sage: g = DynamicalSystem(y^2, B)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Affine Space of dimension 1 over
 Real Field with 53 bits of precision defined by 2 dynamical systems:
  Dynamical System of Affine Space of dimension 1 over
   Real Field with 53 bits of precision
    Defn: Defined on coordinates by sending (x) to (x)
  Dynamical System of Affine Space of dimension 1 over
   Real Field with 53 bits of precision
    Defn: Defined on coordinates by sending (x) to (x^2)

If a dynamical semigroup is built from dynamical systems over number fields, a composite number field is created and all systems will be coerced to it. This composite number field contains all of the initial number fields:

sage: # needs sage.rings.number_field
sage: R.<r> = QQ[]
sage: K.<k> = NumberField(r^2 - 2)
sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: Q.<x,y> = ProjectiveSpace(K, 1)
sage: f = DynamicalSystem([x, y], P)
sage: g = DynamicalSystem([z^2, w^2], Q)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Projective Space of dimension 1 over
 Number Field in k with defining polynomial r^2 - 2 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over
   Number Field in k with defining polynomial r^2 - 2
    Defn: Defined on coordinates by sending (x : y) to (x : y)
  Dynamical System of Projective Space of dimension 1 over
   Number Field in k with defining polynomial r^2 - 2
    Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2)

sage: # needs sage.rings.number_field
sage: R.<r> = QQ[]
sage: K.<k> = NumberField(r^2 - 2)
sage: L.<l> = NumberField(r^2 - 3)
sage: P.<x,y> = ProjectiveSpace(K, 1)
sage: Q.<z,w> = ProjectiveSpace(L, 1)
sage: f = DynamicalSystem([x, y], P)
sage: g = DynamicalSystem([z^2, w^2], Q)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Projective Space of dimension 1 over
 Number Field in kl with defining polynomial r^4 - 10*r^2 + 1
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over
   Number Field in kl with defining polynomial r^4 - 10*r^2 + 1
    Defn: Defined on coordinates by sending (x : y) to (x : y)
  Dynamical System of Projective Space of dimension 1 over
   Number Field in kl with defining polynomial r^4 - 10*r^2 + 1
    Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2)

sage: # needs sage.rings.number_field
sage: R.<r> = QQ[]
sage: K.<k> = NumberField(r^2 - 2)
sage: L.<l> = NumberField(r^2 - 3)
sage: P.<x> = AffineSpace(K, 1)
sage: Q.<y> = AffineSpace(L, 1)
sage: f = DynamicalSystem(x, P)
sage: g = DynamicalSystem(y^2, Q)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Affine Space of dimension 1 over
 Number Field in kl with defining polynomial r^4 - 10*r^2 + 1
 defined by 2 dynamical systems:
  Dynamical System of Affine Space of dimension 1 over
   Number Field in kl with defining polynomial r^4 - 10*r^2 + 1
    Defn: Defined on coordinates by sending (x) to (x)
  Dynamical System of Affine Space of dimension 1 over
   Number Field in kl with defining polynomial r^4 - 10*r^2 + 1
    Defn: Defined on coordinates by sending (x) to (x^2)

A dynamical semigroup may contain dynamical systems over function fields:

sage: R.<r> = QQ[]
sage: P.<x,y> = ProjectiveSpace(R, 1)
sage: f = DynamicalSystem([r * x, y], P)
sage: g = DynamicalSystem([x, r * y], P)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Projective Space of dimension 1 over Univariate
 Polynomial Ring in r over Rational Field defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over
   Univariate Polynomial Ring in r over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (r*x : y)
  Dynamical System of Projective Space of dimension 1 over
   Univariate Polynomial Ring in r over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x : r*y)

sage: R.<r> = QQ[]
sage: P.<x,y> = ProjectiveSpace(R, 1)
sage: f = DynamicalSystem([r * x, y], P)
sage: g = DynamicalSystem([x, y], P)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Projective Space of dimension 1 over Univariate
 Polynomial Ring in r over Rational Field defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over
   Univariate Polynomial Ring in r over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (r*x : y)
  Dynamical System of Projective Space of dimension 1 over
   Univariate Polynomial Ring in r over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x : y)

sage: R.<r,s> = QQ[]
sage: P.<x,y> = ProjectiveSpace(R, 1)
sage: f = DynamicalSystem([r * x, y], P)
sage: g = DynamicalSystem([s * x, y], P)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Projective Space of dimension 1 over Multivariate
 Polynomial Ring in r, s over Rational Field defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over
   Multivariate Polynomial Ring in r, s over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (r*x : y)
  Dynamical System of Projective Space of dimension 1 over
   Multivariate Polynomial Ring in r, s over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (s*x : y)

sage: R.<r,s> = QQ[]
sage: P.<x,y> = ProjectiveSpace(R, 1)
sage: f = DynamicalSystem([r * x, s * y], P)
sage: g = DynamicalSystem([s * x, r * y], P)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Projective Space of dimension 1 over
 Multivariate Polynomial Ring in r, s over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over
   Multivariate Polynomial Ring in r, s over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (r*x : s*y)
  Dynamical System of Projective Space of dimension 1 over
   Multivariate Polynomial Ring in r, s over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (s*x : r*y)

A dynamical semigroup may contain dynamical systems over finite fields:

sage: F = FiniteField(5)
sage: P.<x,y> = ProjectiveSpace(F, 1)
sage: DynamicalSemigroup(([x, y], [x^2, y^2]))
Dynamical semigroup over Projective Space of dimension 1 over
 Finite Field of size 5 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over Finite Field of size 5
    Defn: Defined on coordinates by sending (x : y) to (x : y)
  Dynamical System of Projective Space of dimension 1 over Finite Field of size 5
    Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2)

If a dynamical semigroup is built from dynamical systems over both projective and affine spaces, all systems will be homogenized to dynamical systems over projective space:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: A.<z> = AffineSpace(QQ, 1)
sage: f = DynamicalSystem([x, y], P)
sage: g = DynamicalSystem(z^2, A)
sage: DynamicalSemigroup((f, g))
Dynamical semigroup over Projective Space of dimension 1 over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x : y)
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2)

base_ring()

The base ring of this dynamical semigroup. This is identical to the base ring of all of its defining dynamical system.

OUTPUT: A ring.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x, y], [x^2, y^2]))
sage: f.base_ring()
Rational Field

change_ring(new_ring)

Return a new DynamicalSemigroup whose generators are the initial dynamical systems coerced to new_ring.

INPUT:

• new_ring – a ring.

OUTPUT:

A DynamicalSemigroup defined by this dynamical semigroup’s generators, but coerced to new_ring.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x, y], [x^2, y^2]))
sage: f.change_ring(RR)
Dynamical semigroup over Projective Space of dimension 1 over
 Real Field with 53 bits of precision defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over
   Real Field with 53 bits of precision
    Defn: Defined on coordinates by sending (x : y) to (x : y)
  Dynamical System of Projective Space of dimension 1 over
   Real Field with 53 bits of precision
    Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2)

codomain()

Return the codomain of the generators of this dynamical semigroup.

OUTPUT: A subscheme of a projective space or affine space.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x, y], [x^2, y^2]))
sage: f.codomain()
Projective Space of dimension 1 over Rational Field

defining_polynomials()

Return the set of polynomials that define the generators of this dynamical semigroup.

OUTPUT: A set of polynomials.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x, y], [x^2, y^2]))
sage: f.defining_polynomials()
{(x, y), (x^2, y^2)}

defining_systems()

Return the generators of this dynamical semigroup.

OUTPUT: A tuple of dynamical systems.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x, y], [x^2, y^2]))
sage: f.defining_systems()
(Dynamical System of Projective Space of dimension 1 over Rational Field
   Defn: Defined on coordinates by sending (x : y) to (x : y),
 Dynamical System of Projective Space of dimension 1 over Rational Field
   Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2))

domain()

Return the domain of the generators of this dynamical semigroup.

OUTPUT: A subscheme of a projective space or affine space.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x, y], [x^2, y^2]))
sage: f.domain()
Projective Space of dimension 1 over Rational Field

nth_iterate(p, n)

Return a set of values that results from evaluating this dynamical semigroup on the value p a total of n times.

INPUT:

• p – a value on which dynamical systems can evaluate

• n – a nonnegative integer

OUTPUT: a set of values

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x^2, y^2],))
sage: f.nth_iterate(2, 0)
{(2 : 1)}

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x^2, y^2],))
sage: f.nth_iterate(2, 1)
{(4 : 1)}

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x^2, y^2],))
sage: f.nth_iterate(2, 2)
{(16 : 1)}

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x + y, x - y], [x^2, y^2]))
sage: f.nth_iterate(2, 0)
{(2 : 1)}

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x + y, x - y], [x^2, y^2]))
sage: f.nth_iterate(2, 1)
{(3 : 1), (4 : 1)}

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSemigroup(([x + y, x - y], [x^2, y^2]))
sage: f.nth_iterate(2, 2)
{(5/3 : 1), (2 : 1), (9 : 1), (16 : 1)}

orbit(p, n)

If n is an integer, return \((p, f(p), f^2(p), \dots, f^n(p))\). If n is a list or tuple in interval notation \([a, b]\), return \((f^a(p), \dots, f^b(p))\).

INPUT:

• \(p\) – value on which this dynamical semigroup can be evaluated

• \(n\) – a nonnegative integer or a list or tuple of length 2 describing an interval of the number line containing entirely nonnegative integers

OUTPUT: a tuple of sets of values on the domain of this dynamical semigroup.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: d = DynamicalSemigroup(([x, y], [x^2, y^2]))
sage: d.orbit(2, 0)
({(2 : 1)},)

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: d = DynamicalSemigroup(([x, y], [x^2, y^2]))
sage: d.orbit(2, 1)
({(2 : 1)}, {(2 : 1), (4 : 1)})

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: d = DynamicalSemigroup(([x, y], [x^2, y^2]))
sage: d.orbit(2, 2)
({(2 : 1)}, {(2 : 1), (4 : 1)}, {(2 : 1), (4 : 1), (16 : 1)})

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: d = DynamicalSemigroup(([x, y], [x^2, y^2]))
sage: d.orbit(2, [1, 2])
({(2 : 1), (4 : 1)}, {(2 : 1), (4 : 1), (16 : 1)})

specialization(assignments)

Returns the specialization of the generators of this dynamical semigroup.

INPUT:

• \(assignments\) – argument for specialization of the generators of this dynamical semigroup.

OUTPUT: a dynamical semigroup with the specialization of the generators of this dynamical semigroup.

EXAMPLES:

sage: R.<r> = QQ[]
sage: P.<x,y> = ProjectiveSpace(R, 1)
sage: f = DynamicalSystem([r * x, y], P)
sage: g = DynamicalSystem([x, r * y], P)
sage: d = DynamicalSemigroup((f, g))
sage: d.specialization({r:2})
Dynamical semigroup over Projective Space of dimension 1 over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (2*x : y)
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x : 2*y)

sage: R.<r> = QQ[]
sage: P.<x,y> = ProjectiveSpace(R, 1)
sage: f = DynamicalSystem([r * x, y], P)
sage: g = DynamicalSystem([x, y], P)
sage: d = DynamicalSemigroup((f, g))
sage: d.specialization({r:2})
Dynamical semigroup over Projective Space of dimension 1 over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (2*x : y)
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x : y)

sage: R.<r,s> = QQ[]
sage: P.<x,y> = ProjectiveSpace(R, 1)
sage: f = DynamicalSystem([r * x, y], P)
sage: g = DynamicalSystem([s * x, y], P)
sage: d = DynamicalSemigroup((f, g))
sage: d.specialization({r:2, s:3})
Dynamical semigroup over Projective Space of dimension 1 over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (2*x : y)
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (3*x : y)

sage: R.<r,s> = QQ[]
sage: P.<x,y> = ProjectiveSpace(R, 1)
sage: f = DynamicalSystem([r * x, s * y], P)
sage: g = DynamicalSystem([s * x, r * y], P)
sage: d = DynamicalSemigroup((f, g))
sage: d.specialization({s:3})
Dynamical semigroup over Projective Space of dimension 1 over
 Univariate Polynomial Ring in r over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over
   Univariate Polynomial Ring in r over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (r*x : 3*y)
  Dynamical System of Projective Space of dimension 1 over
   Univariate Polynomial Ring in r over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (3*x : r*y)

class sage.dynamics.arithmetic_dynamics.dynamical_semigroup.DynamicalSemigroup_affine(systems)

Bases: DynamicalSemigroup

A dynamical semigroup defined by multiple dynamical systems on affine space.

INPUT:

• ds_data – list or tuple of dynamical systems or objects that define dynamical systems over affine space.

OUTPUT: DynamicalSemigroup_affine

EXAMPLES:

sage: A.<x> = AffineSpace(QQ, 1)
sage: f = DynamicalSystem(x, A)
sage: g = DynamicalSystem(x^2, A)
sage: DynamicalSemigroup_affine((f, g))
Dynamical semigroup over Affine Space of dimension 1 over
 Rational Field defined by 2 dynamical systems:
  Dynamical System of Affine Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x) to (x)
  Dynamical System of Affine Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x) to (x^2)

homogenize(n)

Return a new DynamicalSemigroup_projective with the homogenization at n of the generators of this dynamical semigroup.

INPUT:

• n – a tuple of nonnegative integers. If n is an integer, then the two values of the tuple are assumed to be the same

OUTPUT: DynamicalSemigroup_projective

EXAMPLES:

sage: A.<x> = AffineSpace(QQ, 1)
sage: f = DynamicalSystem(x + 1, A)
sage: g = DynamicalSystem(x^2, A)
sage: d = DynamicalSemigroup((f, g))
sage: d.homogenize(1)
Dynamical semigroup over Projective Space of dimension 1 over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x0 : x1) to (x0 + x1 : x1)
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x0 : x1) to (x0^2 : x1^2)

sage: A.<x> = AffineSpace(QQ, 1)
sage: f = DynamicalSystem(x + 1, A)
sage: g = DynamicalSystem(x^2, A)
sage: d = DynamicalSemigroup((f, g))
sage: d.homogenize((1, 0))
Dynamical semigroup over Projective Space of dimension 1 over Rational Field
 defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x0 : x1) to (x1 : x0 + x1)
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x0 : x1) to (x1^2 : x0^2)

class sage.dynamics.arithmetic_dynamics.dynamical_semigroup.DynamicalSemigroup_affine_field(systems)

Bases: DynamicalSemigroup_affine

class sage.dynamics.arithmetic_dynamics.dynamical_semigroup.DynamicalSemigroup_affine_finite_field(systems)

Bases: DynamicalSemigroup_affine_field

class sage.dynamics.arithmetic_dynamics.dynamical_semigroup.DynamicalSemigroup_projective(systems)

Bases: DynamicalSemigroup

A dynamical semigroup defined by a multiple dynamical systems on projective space.

INPUT:

• ds_data – list or tuple of dynamical systems or objects that define dynamical systems over projective space.

OUTPUT: DynamicalSemigroup_projective

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: DynamicalSemigroup_projective(([x, y], [x^2, y^2]))
Dynamical semigroup over Projective Space of dimension 1 over
 Rational Field defined by 2 dynamical systems:
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x : y)
  Dynamical System of Projective Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x : y) to (x^2 : y^2)

dehomogenize(n)

Return a new DynamicalSemigroup_projective with the dehomogenization at n of the generators of this dynamical semigroup.

INPUT:

• n – a tuple of nonnegative integers. If n is an integer, then the two values of the tuple are assumed to be the same

OUTPUT: DynamicalSemigroup_affine

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSystem([x, y], P)
sage: g = DynamicalSystem([x^2, y^2], P)
sage: d = DynamicalSemigroup((f, g))
sage: d.dehomogenize(0)
Dynamical semigroup over Affine Space of dimension 1 over
 Rational Field defined by 2 dynamical systems:
  Dynamical System of Affine Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (y) to (y)
  Dynamical System of Affine Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (y) to (y^2)

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSystem([x, y], P)
sage: g = DynamicalSystem([x^2, y^2], P)
sage: d = DynamicalSemigroup((f, g))
sage: d.dehomogenize(1)
Dynamical semigroup over Affine Space of dimension 1 over
 Rational Field defined by 2 dynamical systems:
  Dynamical System of Affine Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x) to (x)
  Dynamical System of Affine Space of dimension 1 over Rational Field
    Defn: Defined on coordinates by sending (x) to (x^2)

class sage.dynamics.arithmetic_dynamics.dynamical_semigroup.DynamicalSemigroup_projective_field(systems)

Bases: DynamicalSemigroup_projective

class sage.dynamics.arithmetic_dynamics.dynamical_semigroup.DynamicalSemigroup_projective_finite_field(systems)

Bases: DynamicalSemigroup_projective_field