Coincidence of invariant measure for the alternate base transformations

Abstract

We characterize all pairs $(\beta ,n),({\beta }^{\prime },m)$ such that the alternate $(\beta ,n)$ and $({\beta }^{\prime },m)$-transformations $K_{(\beta ,n)}$ and $K_{({\beta }^{\prime },m)}$ have the same absolutely continuous invariant measure, where $K_{(\beta ,n)}(i,x)=(i+1\mathrm{mod}2,T_i(x))$ with $i\in \{0,1\}$, $T_0(x)=T_{\beta }(x)=\beta x\mathrm{mod}1$, $T_1(x)=T_n(x)=nx\mathrm{mod}1$ with $\beta >1$ real and $n\ge 2$ an integer.