Fixed point (mathematics)

Representation of a fixed point. This is - according to the criteria given in the text - attractive , i.e. stable .

In mathematics , a fixed point is understood to be a point that is mapped onto itself by a given image. An example: The fixed points of an axis reflection are the points of the mirror axis. A point reflection has only one fixed point, namely its center.

definition

Be a set and a function . Then a point is called a fixed point if it satisfies the equation . ${\displaystyle X}$ ${\displaystyle f\colon X\to X}$ ${\displaystyle x\in X}$ ${\displaystyle f(x)=x}$

Remarks

• If there is a linear mapping on the vector space , then the fixed points of are also called fixed vectors . Since every linear mapping maps the zero vector onto itself, the zero vector is always a fixed vector. If there are other fixed vectors besides the zero vector, then these are eigenvectors of with respect to the eigenvalue 1. ${\displaystyle f\colon X\to X}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle f}$
• For a nonlinear mapping, the associated fixed point equation is an example of a nonlinear equation .
• Every fixed point equation can be rewritten as a zero equation, for example by using. Likewise, every zero equation can be converted into a fixed point equation by z. B. sets. At least in theory, methods for solving one of the two equation forms can also be used for the other. ${\displaystyle f(x)=x}$ ${\displaystyle g(x)=0}$ ${\displaystyle g(x)=f(x)-x}$ ${\displaystyle g(x)=0}$ ${\displaystyle f(x)=x}$ ${\displaystyle f(x)=x+g(x)}$

Fixed points in numerics

In addition, the following applies: The fixed point is stable or unstable if the amount of the derivative of the function under consideration is or is at the point of intersection . This clearly means that the function can be applied to the point itself without changing it, whereby a disturbance changes little (or a lot) by leading to the fixed point (or away from the fixed point). ${\displaystyle \left|f{}'(x)\right|}$ ${\displaystyle <1}$ ${\displaystyle >1}$

Related to the fixed point problem is the problem of "iterated maps", which is important in numerics and chaos research . Starting with a given initial value , one jumps back and forth between the function and the diagonal in a step-like manner , towards the fixed point or away from it, depending on whether the fixed point is stable or unstable. Details can be found in the book by HG Schuster given below. ${\displaystyle x_{1}}$ ${\displaystyle x_{n+1}=f(x_{n})}$ ${\displaystyle f(x)}$

Examples

• The parabolic function given by has two fixed points 0 (stable) and 1 (unstable). ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle f(x)=x^{2}}$
• Let be a vector space and the identical mapping , i.e. the mapping with , then all are fixed points (or fixed vectors). ${\displaystyle V}$ ${\displaystyle \operatorname {Id} \colon V\to V}$ ${\displaystyle \operatorname {Id} x=x}$ ${\displaystyle x\in V}$
• Let the Schwartz space and the continuous Fourier transform . The following applies to the density function of the -dimensional normal distribution . Therefore the density function of the normal distribution is a fixed point of the Fourier transformation. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {F}}\colon {\mathcal {S}}\to {\mathcal {S}}}$ ${\displaystyle \varphi (x)={\tfrac {1}{{\sqrt {2\pi }}^{n}}}\cdot e^{-{\tfrac {1}{2}}x^{2}}}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {F}}(\varphi )=\varphi }$
• The Newton method corresponds to the fixed point equation . ${\displaystyle x_{n+1}=x_{n}-{\tfrac {f(x_{n})}{f'(x_{n})}}}$ ${\displaystyle g(x)=x-{\tfrac {f(x)}{f'(x)}}\,}$

Space with fixed point properties

definition

A topological space has the fixed point property if every continuous mapping has a fixed point. ${\displaystyle X}$ ${\displaystyle f\colon X\to X}$

Examples

• The sphere does not have the fixed point property, because the point reflection at the center point does not have a fixed point. ${\displaystyle S^{n}}$
• A full sphere has the property of fixed points. This is what Brouwer's Fixed Point Theorem says . ${\displaystyle D^{n}}$

Fixed point sets

→ Main article : Fixed point theorem

The existence of fixed points is the subject of some important mathematical theorems. The Banach'sche fixed point theorem states that a contraction of a complete metric space has exactly one fixed point. If a self-mapping is only continuous , the fixed point does not have to be unique and other fixed point sets then only show the existence. In doing so, they usually place stronger requirements on the space in which the function is defined. For example, Schauder's fixed point theorem shows the existence of a fixed point in a compact, convex subset of a Banach space. This theorem is a generalization of Brouwer's Fixed Point Theorem , which states that every continuous mapping of the closed unit sphere has a fixed point in itself. In contrast to the other two theorems, however, this only applies in finite-dimensional spaces, i.e. in or in . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$

Banach's fixed point theorem also provides the convergence and an estimate of the error of the fixed point iteration in the space under consideration. This sentence thus results in a concrete numerical method for calculating fixed points. ${\displaystyle x_{n+1}=f(x_{n})}$

See also

• Fixed straight line
• Fixed point line
• Autonomous differential equation for fixed points in the qualitative theory of ordinary differential equations
• Hyperbolic fixed point

literature

• Vasile I. Instrăţescu: Fixed Point Theory. An Introduction (= Mathematics and its Applications. Vol. 7). D. Reidel, Dordrecht et al. 1981, ISBN 90-277-1224-7 .

Individual evidence

1. ^ Heinz Georg Schuster: Deterministic chaos. An introduction. VCH, Weinheim et al. 1994, ISBN 3-527-29089-3 .
2. ^ Ilka Agricola , Thomas Friedrich : Vector analysis. Differential forms in analysis, geometry and physics. 2nd, revised and expanded edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-1016-8 , p. 36.