A GROUPOID ASSOCIATED TO DISCRETE-TIME SYSTEM THAT DOES NOT SATISFIES SEMIGROUP PROPERTY.

The purpose of this short note is to introduce a groupoid associated to a function f: Xx ▯ → X where X is a uniform space. The motivation is given by the discrete time systems of the form Xn+1 = fn(Xn,,Xn-1,-,X0) that do not satisfy the semigroup property of a process. For these systems the groupoid is induced by the function f defined by f(x,n) := fn(xn-1, xn-2, ...,x0) with X0=x (or ... = x for a fixed n0). The notions of stability and asymptotic stability are introduced in the groupoid framework. [ABSTRACT FROM AUTHOR]