On the fullness of surjective maps of an interval

Source abstract

Let I = [ 0 , 1 ] I = [0,\,1] , B \mathcal {B} = Lebesgue measurable subsets of [ 0 , 1 ] [0,\,1] , and let λ \lambda denote the Lebesgue measure on ( I , B ) (I,\,\mathcal {B}) . Let τ : I → I \tau :I \to I be measurable and surjective. We say τ \tau is full, if for all A ∈ B A \in \mathcal {B} , λ ( A ) > 0 \lambda (A) > 0 , τ ( A ) , τ 2 ( A ) , … \tau (A),\,{\tau ^2}(A), \ldots , measurable, the condition (1) $$lim n → ∞ λ ( τ n ( A ) ) = 1 \lim \limits _{n \to \infty } \lambda ({\tau ^n}(A)) = 1$$ holds. We say τ \tau is interval full if (1) holds for any interval A ⊂ I A \subset I . In this note, we give an example of τ : I → I \tau :I \to I which is continuous and interval full, but not full. We also show that for a class of transformations τ \tau satisfying Renyi’s condition, interval fullness implies fullness. Finally, we show that fullness is not preserved under limits on the surjections.