Generic dynamical systems on schemes¶
This is the generic class for dynamical systems and contains the exported constructor functions. The constructor functions can take either polynomials (or rational functions in the affine case) or morphisms from which to construct a dynamical system. If the domain is not specified, it is constructed. However, if you plan on working with points or subvarieties in the domain, it recommended to specify the domain. For products of projective spaces the domain must be specified.
The initialization checks are always performed by the constructor functions. It is possible, but not recommended, to skip these checks by calling the class initialization directly.
AUTHORS:
 Ben Hutz (July 2017): initial version

class
sage.dynamics.arithmetic_dynamics.generic_ds.
DynamicalSystem
(polys_or_rat_fncts, domain)¶ Bases:
sage.schemes.generic.morphism.SchemeMorphism_polynomial
Base class for dynamical systems of schemes.
INPUT:
polys_or_rat_fncts
– a list of polynomials or rational functions, all of which should have the same parentdomain
– an affine or projective scheme, or product of projective schemes, on whichpolys
defines an endomorphism. Subschemes are also oknames
– (default:('X', 'Y')
) tuple of strings to be used as coordinate names for a projective space that is constructedThe following combinations of
morphism_or_polys
anddomain
are meaningful:morphism_or_polys
is a SchemeMorphism;domain
is ignored in this casemorphism_or_polys
is a list of homogeneous polynomials that define a rational endomorphism ofdomain
morphism_or_polys
is a list of homogeneous polynomials anddomain
is unspecified;domain
is then taken to be the projective space of appropriate dimension over the common parent of the elements inmorphism_or_polys
morphism_or_polys
is a single polynomial or rational function;domain
is ignored and taken to be a 1dimensional projective space over the base ring ofmorphism_or_polys
with coordinate names given bynames
EXAMPLES:
sage: A.<x> = AffineSpace(QQ,1) sage: f = DynamicalSystem_affine([x^2+1]) sage: type(f) <class 'sage.dynamics.arithmetic_dynamics.affine_ds.DynamicalSystem_affine_field'>
sage: P.<x,y> = ProjectiveSpace(QQ,1) sage: f = DynamicalSystem_projective([x^2+y^2, y^2]) sage: type(f) <class 'sage.dynamics.arithmetic_dynamics.projective_ds.DynamicalSystem_projective_field'>
sage: P1.<x,y> = ProjectiveSpace(CC,1) sage: H = End(P1) sage: DynamicalSystem(H([y, x])) Dynamical System of Projective Space of dimension 1 over Complex Field with 53 bits of precision Defn: Defined on coordinates by sending (x : y) to (y : x)
DynamicalSystem
defaults to projective:sage: R.<x,y,z> = QQ[] sage: DynamicalSystem([x^2, y^2, z^2]) Dynamical System of Projective Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x : y : z) to (x^2 : y^2 : z^2)
sage: A.<x,y> = AffineSpace(QQ, 2) sage: DynamicalSystem([y, x], domain=A) Dynamical System of Affine Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x, y) to (y, x) sage: H = End(A) sage: DynamicalSystem(H([y, x])) Dynamical System of Affine Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x, y) to (y, x)
Note that
domain
is ignored if an endomorphism is passed in:sage: P.<x,y> = ProjectiveSpace(QQ, 1) sage: P2.<x,y> = ProjectiveSpace(CC, 1) sage: H = End(P2) sage: f = H([CC.0*x^2, y^2]) sage: g = DynamicalSystem(f, domain=P) sage: g.domain() Projective Space of dimension 1 over Complex Field with 53 bits of precision
Constructing a common parent:
sage: P.<x,y> = ProjectiveSpace(ZZ, 1) sage: DynamicalSystem([CC.0*x^2, 4/5*y^2]) Dynamical System of Projective Space of dimension 1 over Complex Field with 53 bits of precision Defn: Defined on coordinates by sending (x : y) to (1.00000000000000*I*x^2 : 0.800000000000000*y^2) sage: P.<x,y> = ProjectiveSpace(GF(5), 1) sage: K.<t> = GF(25) sage: DynamicalSystem([GF(5)(3)*x^2, K(t)*y^2]) Dynamical System of Projective Space of dimension 1 over Finite Field in t of size 5^2 Defn: Defined on coordinates by sending (x : y) to (2*x^2 : (t)*y^2)

as_scheme_morphism
()¶ Return this dynamical system as
SchemeMorphism_polynomial
.OUTPUT:
SchemeMorphism_polynomial
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(ZZ, 2) sage: f = DynamicalSystem_projective([x^2, y^2, z^2]) sage: type(f.as_scheme_morphism()) <class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space'>
sage: P.<x,y> = ProjectiveSpace(QQ, 1) sage: f = DynamicalSystem_projective([x^2y^2, y^2]) sage: type(f.as_scheme_morphism()) <class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_field'>
sage: P.<x,y> = ProjectiveSpace(GF(5), 1) sage: f = DynamicalSystem_projective([x^2, y^2]) sage: type(f.as_scheme_morphism()) <class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_finite_field'>
sage: A.<x,y> = AffineSpace(ZZ, 2) sage: f = DynamicalSystem_affine([x^22, y^2]) sage: type(f.as_scheme_morphism()) <class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space'>
sage: A.<x,y> = AffineSpace(QQ, 2) sage: f = DynamicalSystem_affine([x^22, y^2]) sage: type(f.as_scheme_morphism()) <class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_field'>
sage: A.<x,y> = AffineSpace(GF(3), 2) sage: f = DynamicalSystem_affine([x^22, y^2]) sage: type(f.as_scheme_morphism()) <class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_finite_field'>

change_ring
(R, check=True)¶ Return a new dynamical system which is this map coerced to
R
.If
check
isTrue
, then the initialization checks are performed.INPUT:
R
– ring or morphism
OUTPUT:
A new
DynamicalSystem_projective
that is this map coerced toR
.EXAMPLES:
sage: P.<x,y> = ProjectiveSpace(ZZ, 1) sage: f = DynamicalSystem_projective([3*x^2, y^2]) sage: f.change_ring(GF(5)) Dynamical System of Projective Space of dimension 1 over Finite Field of size 5 Defn: Defined on coordinates by sending (x : y) to (2*x^2 : y^2)

field_of_definition_critical
(return_embedding=False, simplify_all=False, names='a')¶ Return smallest extension of the base field which contains the critical points
Ambient space of dynamical system must be either the affine line or projective line over a number field or finite field.
INPUT:
return_embedding
– (default:False
) boolean; IfTrue
, return an embedding of base field of dynamical system into the returned number field or finite field. Note that computing this embedding might be expensive.simplify_all
– (default:False
) boolean; IfTrue
, simplify intermediate fields and also the resulting number field. Note that this is not implemented for finite fields and has no effectnames
– (optional) string to be used as generator for returned number field or finite field
OUTPUT:
If
return_embedding
isFalse
, the field of definition as an absolute number field or finite field. Ifreturn_embedding
isTrue
, a tuple(K, phi)
wherephi
is an embedding of the base field inK
.EXAMPLES:
Note that the number of critical points is 2d2, but (1:0) has multiplicity 2 in this case:
sage: P.<x,y> = ProjectiveSpace(QQ, 1) sage: f = DynamicalSystem([1/3*x^3 + x*y^2, y^3], domain=P) sage: f.critical_points() [(1 : 0)] sage: N.<a> = f.field_of_definition_critical(); N Number Field in a with defining polynomial x^2 + 1 sage: g = f.change_ring(N) sage: g.critical_points() [(a : 1), (a : 1), (1 : 0)]
sage: A.<z> = AffineSpace(QQ, 1) sage: f = DynamicalSystem([z^4 + 2*z^2 + 2], domain=A) sage: K.<a> = f.field_of_definition_critical(); K Number Field in a with defining polynomial z^2 + 1
sage: G.<a> = GF(9) sage: R.<z> = G[] sage: R.irreducible_element(3, algorithm='first_lexicographic') z^3 + (a + 1)*z + a sage: A.<x> = AffineSpace(G,1) sage: f = DynamicalSystem([x^4 + (2*a+2)*x^2 + a*x], domain=A) sage: f[0].derivative(x).univariate_polynomial().is_irreducible() True sage: f.field_of_definition_critical(return_embedding=True, names='b') (Finite Field in b of size 3^6, Ring morphism: From: Finite Field in a of size 3^2 To: Finite Field in b of size 3^6 Defn: a > 2*b^5 + 2*b^3 + b^2 + 2*b + 2)

field_of_definition_periodic
(n, formal=False, return_embedding=False, simplify_all=False, names='a')¶ Return smallest extension of the base field which contains all fixed points of the
n
th iterateAmbient space of dynamical system must be either the affine line or projective line over a number field or finite field.
INPUT:
n
– a positive integerformal
– (default:False
) boolean;True
signals to return number field or finite field over which the formal periodic points are defined, where a formal periodic point is a root of then
th dynatomic polynomial.False
specifies to find number field or finite field over which all periodic points of then
th iterate are definedreturn_embedding
– (default:False
) boolean; IfTrue
, return an embedding of base field of dynamical system into the returned number field or finite field. Note that computing this embedding might be expensive.simplify_all
– (default:False
) boolean; IfTrue
, simplify intermediate fields and also the resulting number field. Note that this is not implemented for finite fields and has no effectnames
– (optional) string to be used as generator for returned number field or finite field
OUTPUT:
If
return_embedding
isFalse
, the field of definition as an absolute number field or finite field. Ifreturn_embedding
isTrue
, a tuple(K, phi)
wherephi
is an embedding of the base field inK
.EXAMPLES:
sage: P.<x,y> = ProjectiveSpace(QQ, 1) sage: f = DynamicalSystem([x^2, y^2], domain=P) sage: f.periodic_points(3, minimal=False) [(0 : 1), (1 : 0), (1 : 1)] sage: N.<a> = f.field_of_definition_periodic(3); N Number Field in a with defining polynomial x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 sage: f.periodic_points(3,minimal=False, R=N) [(0 : 1), (a : 1), (a^5 : 1), (a^2 : 1), (a^5  a^4  a^3  a^2  a  1 : 1), (a^4 : 1), (1 : 0), (a^3 : 1), (1 : 1)]
sage: A.<z> = AffineSpace(QQ, 1) sage: f = DynamicalSystem([(z^2 + 1)/(2*z + 1)], domain=A) sage: K.<a> = f.field_of_definition_periodic(2); K Number Field in a with defining polynomial z^4 + 12*z^3 + 39*z^2 + 18*z + 171 sage: F.<b> = f.field_of_definition_periodic(2, formal=True); F Number Field in b with defining polynomial z^2 + 3*z + 6
sage: G.<a> = GF(4) sage: A.<x> = AffineSpace(G, 1) sage: f = DynamicalSystem([x^2 + (a+1)*x + 1], domain=A) sage: g = f.nth_iterate_map(2)[0] sage: (gx).univariate_polynomial().factor() (x + 1) * (x + a + 1) * (x^2 + a*x + 1) sage: f.field_of_definition_periodic(2, return_embedding=True, names='b') (Finite Field in b of size 2^4, Ring morphism: From: Finite Field in a of size 2^2 To: Finite Field in b of size 2^4 Defn: a > b^2 + b)

field_of_definition_preimage
(point, n, return_embedding=False, simplify_all=False, names='a')¶ Return smallest extension of the base field which contains the
n
th preimages ofpoint
Ambient space of dynamical system must be either the affine line or projective line over a number field or finite field.
INPUT:
point
– a point in this map’s domainn
– a positive integerreturn_embedding
– (default:False
) boolean; IfTrue
, return an embedding of base field of dynamical system into the returned number field or finite field. Note that computing this embedding might be expensive.simplify_all
– (default:False
) boolean; IfTrue
, simplify intermediate fields and also the resulting number field. Note that this is not implemented for finite fields and has no effectnames
– (optional) string to be used as generator for returned number field or finite field
OUTPUT:
If
return_embedding
isFalse
, the field of definition as an absolute number field or finite field. Ifreturn_embedding
isTrue
, a tuple(K, phi)
wherephi
is an embedding of the base field inK
.EXAMPLES:
sage: P.<x,y> = ProjectiveSpace(QQ, 1) sage: f = DynamicalSystem([1/3*x^2 + 2/3*x*y, x^2  2*y^2], domain=P) sage: N.<a> = f.field_of_definition_preimage(P(1,1), 2, simplify_all=True); N Number Field in a with defining polynomial x^8  4*x^7  128*x^6 + 398*x^5 + 3913*x^4  8494*x^3  26250*x^2 + 30564*x  2916
sage: A.<z> = AffineSpace(QQ, 1) sage: f = DynamicalSystem([z^2], domain=A) sage: K.<a> = f.field_of_definition_preimage(A(1), 3); K Number Field in a with defining polynomial z^4 + 1
sage: G = GF(5) sage: P.<x,y> = ProjectiveSpace(G, 1) sage: f = DynamicalSystem([x^2 + 2*y^2, y^2], domain=P) sage: f.field_of_definition_preimage(P(2,1), 2, return_embedding=True, names='a') (Finite Field in a of size 5^2, Ring morphism: From: Finite Field of size 5 To: Finite Field in a of size 5^2 Defn: 1 > 1)

specialization
(D=None, phi=None, homset=None)¶ Specialization of this dynamical system.
Given a family of maps defined over a polynomial ring. A specialization is a particular member of that family. The specialization can be specified either by a dictionary or a
SpecializationMorphism
.INPUT:
D
– (optional) dictionaryphi
– (optional) SpecializationMorphismhomset
– (optional) homset of specialized map
OUTPUT:
DynamicalSystem
EXAMPLES:
sage: R.<c> = PolynomialRing(QQ) sage: P.<x,y> = ProjectiveSpace(R, 1) sage: f = DynamicalSystem_projective([x^2 + c*y^2,y^2], domain=P) sage: f.specialization({c:1}) Dynamical System of Projective Space of dimension 1 over Rational Field Defn: Defined on coordinates by sending (x : y) to (x^2 + y^2 : y^2)