Uniform structure with iterated function system, step skew product and their uniform entropy

Let F : Sigma(+)(m) x X -> Sigma(+)(m) x X by (omega,x) 7 -> (sigma(m)(omega), f(omega 0) ( x )), be skew product of IFS F = {f(i) : X -> X |i = 0,1,. . . , m - 1} on uniform space (X, U-X). In this paper, we prove that the equivalence of the chain transitive, topologically transitive and topological shadowing property between IFS F and step skew product F . Moreover, we give two version of entropy, uniform entropy and uniform covering entropy, for IFS F on uniform space (X, U-X), and prove that the basic properties of them. Finally, we show that h(u)(F) = log m + h(u)(F), where hu is uniform entropy.