Fixed point index
Encyclopedia
In mathematics
, the fixed-point index is a concept in topological fixed-point
theory, and in particular Nielsen theory
. The fixed-point index can be thought of as a multiplicity
measurement for fixed points.
The index can be easily defined in the setting of complex analysis
: Let f(z) be a holomorphic mapping on the complex plane, and let z0 be a fixed point of f. Then the function f(z) − z is holomorphic, and has an isolated zero at z0. We define the fixed point index of f at z0, denoted i(f, z0), to be the multiplicity of the zero of the function f(z) − z at the point z0.
In real Euclidean space, the fixed-point index is defined as follows: If x0 is an isolated fixed point of f, then let g be the function defined by
Then g has an isolated singularity at x0, and maps the boundary of some deleted neighborhood of x0 to the unit sphere. We define i(f, x0) to be the Brouwer degree
of the mapping induced by g on some suitably chosen small sphere around x0.
–Hopf
theorem, which states:
where Fix(f) is the set of fixed points of f, and Λf is the Lefschetz number of f.
Since the quantity on the left-hand side of the above is clearly zero when f has no fixed points, the Lefschetz–Hopf theorem trivially implies the Lefschetz fixed point theorem.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the fixed-point index is a concept in topological fixed-point
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
theory, and in particular Nielsen theory
Nielsen theory
Nielsen theory is a branch of mathematical research with its origins in topological fixed point theory. Its central ideas were developed by Danish mathematician Jakob Nielsen, and bear his name....
. The fixed-point index can be thought of as a multiplicity
Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point....
measurement for fixed points.
The index can be easily defined in the setting of complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
: Let f(z) be a holomorphic mapping on the complex plane, and let z0 be a fixed point of f. Then the function f(z) − z is holomorphic, and has an isolated zero at z0. We define the fixed point index of f at z0, denoted i(f, z0), to be the multiplicity of the zero of the function f(z) − z at the point z0.
In real Euclidean space, the fixed-point index is defined as follows: If x0 is an isolated fixed point of f, then let g be the function defined by
Then g has an isolated singularity at x0, and maps the boundary of some deleted neighborhood of x0 to the unit sphere. We define i(f, x0) to be the Brouwer degree
Degree of a continuous mapping
In topology, the degree is a numerical invariant that describes a continuous mapping between two compact oriented manifolds of the same dimension. Intuitively, the degree represents the number of times that the domain manifold wraps around the range manifold under the mapping...
of the mapping induced by g on some suitably chosen small sphere around x0.
The Lefschetz–Hopf theorem
The importance of the fixed-point index is largely due to its role in the LefschetzSolomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
–Hopf
Heinz Hopf
Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...
theorem, which states:
where Fix(f) is the set of fixed points of f, and Λf is the Lefschetz number of f.
Since the quantity on the left-hand side of the above is clearly zero when f has no fixed points, the Lefschetz–Hopf theorem trivially implies the Lefschetz fixed point theorem.