By Theorem 2.3, an affine map (d,D) [member of] aff (G) exists which also satisfies [~.f] *([gamma]) [??] (d,D) = (d,D) [??] [gamma] for all [gamma] [member of] [GAMMA].

The map (d, D) will be called an affine homotopy lift of f, while we will denote the map [bar.(d, D)] as an affine map on an infra-nilmanifold.

Suppose that g is the affine map on [GAMMA]\G that is induced by an affine homotopy lift [~.g] of f.

Instead one starts with an easily invertible quadratic map F (called central map) and combines it with two invertible

affine maps S and T to get a public key of the form P = S * F * T .

The first one we use to illustrate the meaning of the notions linear and affine map and to explain the role of the matrix of a linear transformation.

An affine map X' = M x X + T is a composition of a linear map, given by the matrix M and a translation, given by the matrix (or a column vector) T.

Let [[beta].sub.i] [member of] R, [b.sub.i] [member of] [R.sup.n] for i [member of] I and E: [R.sup.n] [right arrow] [R.sup.n] be an idempotent affine map. Then the set {x [member of] [R.sup.n]: (E(x), [b.sub.i]) [rho] [[beta].sub.i] for each i [member of] I} is E-convex where [rho] [member of]{<, =, >, [less than or equal to], [greater than or equal to]}.

Let E: [R.sup.n] [right arrow] [R.sup.n] be an idempotent affine map. Let C be an E-convex subset of [R.sup.n].

It is the inverse problem of Definition 10 and can be easily computed by using the affine map in Definition 10.

In Remark 11, the way of computing the constraint set of variable x(k | k) by affine map is shown.

[46], Corollary on p.50.) Moreover, C is invariant under the action of [??] by

affine maps. But every compact group admits a fixed point in every convex compact set upon which it acts by affine homeomorphisms (this is a reformulation of a statement that every compact group is amenable).

Theorem 1.11 of [22] says that if X is a connected, metric, compact abelian group, y is its normalized Haar measure and V is an affine map, then V being ergodic is equivalent to having a point with dense orbit, in which case the set of points with dense orbit has measure 1.

Then the affine map L = aB is also ergodic and in particular AL is ergodic for all [lambda] [member of] T.

A set of TV one-to-one contraction

affine maps [w.sub.i]: X [right arrow] X, [W.sub.i](x) = [S.sub.i]x + [a.sub.i], with the condition that [[union].sup.N.sub.i=1][W.sub.i](X) = X,

A simple example in this class is the self-affine set generated by the

affine maps