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Orbit (mathematics)
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Orbit (mathematics)

In mathematics, an orbit is a concept in group theory. Consider a group G acting on a set X. The orbit of an element x of X is the set of elements of X to which x can be moved by the elements of G; it is denoted by Gx. That is

The orbits of a group action are the equivalence classes of the equivalence relation on X defined by x ~ y iff there exists g in G with x = g.y. As a consequence, every element of X belongs to one and only one orbit.

If two elements x and y belong to the same orbit, then their stabilizer subgroups Gx and Gy are isomorphic. More precisely: if y = g.x, then the inner automorphism of G given by h |-> ghg-1 maps Gx to Gy.

If both G and X are finite, then the size of any orbit is a factor of the order of the group G by the orbit-stabilizer theorem.

The set of all orbits is denoted by X/G. Burnside's lemma gives a formula that allows to calculate the number of orbits.

See also:

In the study of dynamical systems, an orbit is the sequence generated by iterating a map.

An orbit is called closed if this sequence is finite. In simple terms, this means that the orbit will repeat itself. Such an orbit may be periodic, meaning that the entire sequence repeats. Othewise it is eventually periodic, meaning that the sequence will start in a non-repeating orbit but will enter a repeating orbit after some finite number of iterations.

If the map is on a metric space, an orbit is asymptotically periodic if the orbit converges to a periodic orbit.

The most interesting orbits are those that are chaotic. These orbits are not closed or asymptotically periodic. They also demonstrate sensitive dependence on initial conditions, meaning that small differences in the starting value will cause large differences in the subsequent orbits.