Sturmian expansions and entropy (Q2855530)

The author proves that for any \(\alpha \in [0,1)\setminus\mathbb{Q}\) the correspond Sturmian \(\alpha\)-expansions are valid. He further generalize this to the case of \(n\)-interval exchanges maps by proving that for any irreducible interval exchanges transformation (IET) with rationally independent interval lengths, the corresponding IET-expansions are valid.NEWLINENEWLINEWe remind that for any monotone function \(f\) with values in \([0,1]\) the corresponding \(f\)-expansion is an expression of the form NEWLINE\[NEWLINEx=f(d_1+f(d_2+f(d_3+\dots))),NEWLINE\]NEWLINE where the digits \(d_k\) are integers. It is turn out thatNEWLINENEWLINE\(\bullet\) if \(f(x)=\frac{1}{x}\) then the \(f\)-expansion is the well-known continued fraction expansion,NEWLINENEWLINE\(\bullet\) and if \(f(x)=\frac{x}{\beta}\) then the \(f\)-expansion is the \(\beta\)-expansion.NEWLINENEWLINEFinally, for Sturmian \(\alpha\)-expansions, the correspond function \(f\) is given by NEWLINE\[NEWLINEf(x)=\begin{cases} 0, & \text{if \(x < \alpha\);} \\ x-\alpha, & \text{if \(\alpha \leq x \leq \alpha+1\);} \\ 1, & \text{if \(x> \alpha+1\).} \end{cases} NEWLINE\]