The baker's map can be defined as $E_{2}:S^{1}\rightarrow S^{1}$, $E_{2}x=2x \mod 1$. This map has various properties, for example, denoting Lebesgue measure by $\lambda$, we have $\lambda(E_{2}^{-1}(A))=\lambda(A)$ for any measurable subset $A\subset S^{1}$, and the set of points with dense orbits under $E_{2}$ has Lebesgue measure 1.
Define a map $\Phi=\phi\circ E_{2}$, where $\phi:[0,1]\rightarrow[0,1]$ is continuous and strictly increasing with $\phi(0)=0,\phi(1)=1$. Assume also that $\phi'$ is continuous. Are there properties similar to those above for the map $\Phi$? [I'm more interested in the second property in particular since I think the first isn't valid.]