ИСТИНА

Consider a finitely generated semigroup of affine operators SS with generators A1,…,AkA1,…,Ak acting in Euclidean nn-dimensional space. We call it bounded if an orbit of an arbitrary point under the action of the semigroup is bounded. Assume SS is bounded. The central question we consider is the asymptotics of "long" products of the generating operators A1,…,AkA1,…,Ak. More precisely, we take a random infinite tuple of indices i1,i2,i3,…i1,i2,i3,… where is∈{1,…,k}is∈{1,…,k}. Consider an associated sequence of operators Ai1,(Ai2Ai1),(Ai3Ai2Ai1),…Ai1,(Ai2Ai1),(Ai3Ai2Ai1),…. When do these operators stabilise an arbitrary point p∈Rnp∈Rn? That is, when the sequence of the images Ai1p,(Ai2Ai1)p,…Ai1p,(Ai2Ai1)p,… converges? The techniques we apply to investigate these questions is functional equations. There is a certain family of functional equations that involves the generators of the semigroup SS, so called self-similarity equations. It turns out that the existence of a solution of these equations is closely related to an asymptotic behaviour of the semigroup SS. Applying results on these self-similarity equations we study the structure of the semigroup SS. This is a joint work with V. Protasov.