On Growth Rates Of Continuous Functions In Skew-product System

Let X and Y be compact topological spaces and f: X→X be a continuous trans-formation. Let F: X×Y→X×Y be a continuous transformation such that F(x,y)=(f(x), g(x, y)) where g: X×Y→Y is a continuous transformation and letÏ€: X×Y→Xbe the project map. Given a continuous functionψ: X×Y→R. Let M(F) andε(f)denote the set of all F-invariant Borel probability measures on X×Y and the set of allergodic f-invariant measures on X respectively. For any f-invariant measureμ, we denoteM_μ(F)={ν∈M(F):Ï€^*ν=μ}.If f-invariant measureμis ergodic, we define the maximal growth rate ofψto beΛ(μ),It is shown that it is equal toλ(μ), whereΛ(μ)=max₍ν∈M_μ(F)∫_X×Yψdν, (?)μ∈ε(f), andλ(μ)=lim_{n→∞}1／n max_y∈YΣ_i=0^n-1ψ(Fⁱ(x,y))=const, forμa.e.x.If f-invariant measureμis not ergodic, we study the ergodic component ofμ, and weintroduce the definition of essential supreme. We define the maximal essential growth rateofψto be (?)-esssup_{(?)∈ε(f)}Λ((?)). It is shown that it is equal toμ-esssup_x∈Xλ(x), whereμ-esssup A denotes the essential supreme of A,Λ((?))=max_{ν∈M(?)(F)}∫_X×Yψdν,(?)(?)∈ε(f),andλ(x)=lim_{n→∞}1／n max_y∈YΣ_i=0^n-1ψ(Fⁱ(x, y)) forμa.e.x. Here (?) is the ergodic componentofμ.