Let $X$ be set. Let $f$ be a function from $X$ into $X$. For a given set $E\subseteq X$, we say $E$ determines a $U$-part of $f$ if $f(E)\subseteq E$ and the restriction $f:E\to E$ is a bijection. 


I am looking for a commuting pair of ***injective*** functions $f:X\to X$ and $g:X\to X$ (I mean $fg=gf$) such that both $f$ and $g$ have $U$-parts but not common $U$-part. 



A commuting pair of functions with no common bijective part
Let $X=\mathbb{N}$, with
$f(2n)=2n$, $f(2n+1)=2n+3$ and $g(2n)=2n+2$, $g(2n+1)=2n+1$.


