Let $T: (0,1)\setminus\mathbb Q \to (0,1)\setminus\mathbb Q$ be the Gauss map defined by $T(x)=1/x-\lfloor 1/x\rfloor$, where $\lfloor\cdot\rfloor: \mathbb R\to\mathbb Z$ denotes the floor function. For each integer $k\geq 1$, set
If $x\in I_k\setminus\mathbb Q$, then the next partial quotient of $x=[0;a_1,a_2,\dots]$ is $a_1=k$, and $T$ maps $I_k\setminus\mathbb Q$ bijectively onto $(0,1)\setminus\mathbb Q$. On $I_k$, the inverse branch is