Discrete fixed-point theorem

In discrete mathematics, a discrete fixed-point theorem is a fixed-point theorem for functions defined on finite sets, typically subsets of the integer grid $\mathbb {Z} ^{n}$.

Discrete fixed-point theorems were developed by Iimura, Murota and Tamura, Yang, Chen and Deng and others.

Basic concepts

Continuous fixed-point theorems often require a continuous function. Since continuity is not meaningful for functions on discrete sets, it is replaced by conditions such as a direction-preserving function. Such conditions imply that the function does not change too drastically when moving between neighboring points of the integer grid.

Continuous fixed-point theorems often require a convex set. The analogue of this property for discrete sets is an integrally-convex set.

For functions on finite sets

Given a set X in $\mathbb {Z} ^{n}$, we denote by ch(X) its convex hull, which is a subset of $\mathbb {R} ^{n}$.

Iimura-Murota-Tamura theorem: Let X be a finite integrally-convex subset of $\mathbb {Z} ^{n}$. Let f: X → ch(X) be a hypercubic direction-preserving (HDP) function. Then f has a fixed-point in X.

Chen-Deng theorem: Let X be a finite set contained in $\mathbb {R} ^{n}$. Let f: X → ch(X) be a simplicially direction-preserving (SDP) function. Then f has a fixed-point in X.

For discontinuous functions on infinite sets

Discrete fixed-point theorems are closely related to fixed-point theorems on discontinuous functions. These, too, use the direction-preservation condition instead of continuity.

Herings-Laan-Talman-Yang fixed-point theorem: Let X be a non-empty polytope in $\mathbb {R} ^{n}$. Let f: X → X be a locally gross direction preserving (LGDP) function: at any point x that is not a fixed point of f, the direction of $f(x)-x$ is grossly preserved in some neighborhood of x, in the sense that for any two points y, z in this neighborhood, its inner product is non-negative, i.e.: $(f(y)-y)\cdot (f(z)-z)\geq 0$. Note that every continuous function is LGDP, but an LGDP function may be discontinuous. An LGDP function may even be neither upper nor lower semi-continuous. Then f has a fixed point in X. Moreover, there is a constructive algorithm for approximating this fixed point.

See also

• Sperner's lemma - a fixed-point-like lemma on vertices of a triangulation.
• Knaster–Tarski theorem and Bourbaki–Witt theorem - two fixed-point theorems on partially-ordered sets.