A-equivalence

In mathematics, formula_1-equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs.
Let formula_2 and formula_3 be two manifolds, and let formula_4 be two smooth map germs. We say that formula_5 and formula_6 are formula_1-equivalent if there exist diffeomorphism germs formula_8 and formula_9 such that formula_10
In other words, two map germs are formula_1-equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. formula_2) and the target (i.e. formula_3).
Let formula_14 denote the space of smooth map germs formula_15 Let formula_16 be the group of diffeomorphism germs formula_17 and 
formula_18 be the group of diffeomorphism germs formula_19 The group formula_20 acts on formula_14 in the natural way: formula_22 Under this action we see that the map germs formula_4 are formula_1-equivalent if, and only if, formula_6 lies in the orbit of formula_5, i.e. formula_27 (or vice versa).
A map germ is called stable if its orbit under the action of formula_20 is open relative to the Whitney topology. Since formula_14 is an infinite dimensional space metric topology is no longer trivial. Whitney topology compares the differences in successive derivatives and gives a notion of proximity within the infinite dimensional space. A base for the open sets of the topology in question is given by taking formula_30-jets for every formula_30 and taking open neighbourhoods in the ordinary Euclidean sense. Open sets in the topology are then unions of
these base sets.
Consider the orbit of some map germ formula_32 The map germ formula_5 is called simple if there are only finitely many other orbits in a neighbourhood of each of its points. Vladimir Arnold has shown that the only simple singular map germs formula_34 for formula_35 are the infinite sequence formula_36 (formula_37), the infinite sequence formula_38 (formula_37), formula_40 formula_41 and formula_42