Fundamental Domain - Hints at General Definition

Hints At General Definition

Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain (also called fundamental region) for this action is a set D of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several precisely defined ways. One typical condition is that D is almost an open set, in the sense that D is the symmetric difference of an open set in G with a set of measure zero, for a certain (quasi)invariant measure on X. A fundamental domain always contains a free regular set U, an open set moved around by G into disjoint copies, and nearly as good as D in representing the orbits. Frequently D is required to be a complete set of coset representatives with some repetitions, but the repeated part has measure zero. This is a typical situation in ergodic theory. If a fundamental domain is used to calculate an integral on X/G, sets of measure zero do not matter.

For example, when X is Euclidean space Rn of dimension n, and G is the lattice Zn acting on it by translations, the quotient X/G is the n-dimensional torus. A fundamental domain D here can be taken to be n, whose boundary consists of the points whose orbit has more than one representative in D.

Read more about this topic:  Fundamental Domain

Famous quotes containing the words hints, general and/or definition:

    If we will admit time into our thoughts at all, the mythologies, those vestiges of ancient poems, wrecks of poems, so to speak, the world’s inheritance,... these are the materials and hints for a history of the rise and progress of the race; how, from the condition of ants, it arrived at the condition of men, and arts were gradually invented. Let a thousand surmises shed some light on this story.
    Henry David Thoreau (1817–1862)

    According to the historian, they escaped as by a miracle all roving bands of Indians, and reached their homes in safety, with their trophies, for which the General Court paid them fifty pounds. The family of Hannah Dustan all assembled alive once more, except the infant whose brains were dashed out against the apple tree, and there have been many who in later time have lived to say that they have eaten of the fruit of that apple tree.
    Henry David Thoreau (1817–1862)

    The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.
    Samuel Taylor Coleridge (1772–1834)