Typical properties of interval maps preserving the Lebesgue measure

Abstract

Let us denote $\lambda$ the Lebesgue measure on $[0,1]$, put$$C(\lambda )=\{f\in C([0,1]);\forall A\subset [0,1],A\text{Borel}:\lambda (A)=\lambda (f^{-1}(A))\}.$$ We endow the set $C(\lambda )$ by the uniform metric $\rho$ and investigate dynamical properties of typical maps in the complete metric space $(C(\lambda ),\rho )$.