Generic property

In mathematics, properties of objects are called generic if they are typical in a certain way and only inapplicable in special pathological cases. There is a mathematically clearly defined use of the term “generic”. In addition, the term is also used informally to express that a property “mostly” or “almost always” applies.

One often speaks of generic properties of functions or vector fields , for example in the singularity theory or in the theory of ordinary differential equations and dynamic systems . In this case one considers the function as an element of a function space and means that the corresponding property is generic for elements of this function space.

definition

Let it be a topological space , for example a function space . A property (of elements of ) is called generic if the set of satisfying elements is a residual set , i.e. a countable average of open, dense subsets of . ${\ displaystyle X}$ ${\ displaystyle P}$ ${\ displaystyle X}$ ${\ displaystyle P}$ ${\ displaystyle X}$

One then also says: a generic element of has the property . ${\ displaystyle X}$ ${\ displaystyle P}$

Examples

A generic real number is a Liouville number .

A generic real number is irrational . On the other hand, rationality is not a generic property, although the rational numbers are close to the real numbers.

The coordinates of a generic point are algebraically independent . ${\ displaystyle x \ in \ mathbb {R} ^ {n}}$

A generic differentiable function is a Morse function . ${\ displaystyle f \ colon M \ to \ mathbb {R}}$

A generic diffeomorphism of a compact manifold has only hyperbolic periodic points . Furthermore, the periodic points of a generic diffeomorphism lie close to . ${\ displaystyle f \ colon M \ to M}$ ${\ displaystyle M}$

A later disproved conjecture by Smale was that the flow of a generic vector field satisfies Axiom A.

A generic point of an algebraic variety over a field of characteristic zero is smooth .

Function rooms

In the space of the functions between two manifolds and every residual subset is dense, which is why functions with a generic property are always dense in the function space . ${\ displaystyle C ^ {r} (M, N)}$ ${\ displaystyle C ^ {r}}$ ${\ displaystyle M}$ ${\ displaystyle N}$

Algebraic Geometry

Regarding the Zariski topology on irreducible algebraic varieties , a non-empty open set is always dense, so the definition of a generic property can be reformulated there as follows: a property is generic if it applies to a non-empty Zariski open subset .

literature

Stephen Smale : Differentiable dynamical systems . Bull. Amer. Math. Soc. Volume 73, Number 6 (1967), 747-817. pdf
René Thom : Les singularités des applications différentiables. Ann. Inst. Fourier, Grenoble 6 (1955-1956), 43-87. pdf