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 parent

• domain – an affine or projective scheme, or product of projective schemes, on which polys defines an endomorphism. Subschemes are also ok

• names – (default: ('X', 'Y')) tuple of strings to be used as coordinate names for a projective space that is constructed

  The following combinations of morphism_or_polys and domain are meaningful:

  • morphism_or_polys is a SchemeMorphism; domain is ignored in this case
  • morphism_or_polys is a list of homogeneous polynomials that define a rational endomorphism of domain
  • morphism_or_polys is a list of homogeneous polynomials and domain is unspecified; domain is then taken to be the projective space of appropriate dimension over the base ring of the first element of morphism_or_polys
  • morphism_or_polys is a single polynomial or rational function; domain is ignored and taken to be a 1-dimensional projective space over the base ring of morphism_or_polys with coordinate names given by names

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)

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^2-y^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^2-2, 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^2-2, 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^2-2, 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 is True, then the initialization checks are performed.

INPUT:

• R – ring or morphism

OUTPUT:

A new DynamicalSystem_projective that is this map coerced to R.

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)

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) dictionary
• phi – (optional) SpecializationMorphism
• homset – (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)