Coincidence of invariant measure for the alternate base transformations

Abstract

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

Figures (5)

• Figure 1: The map $K_{(\beta_0,\ldots,\beta_{p-1})}$.
• Figure 2: Top: $\mathcal{I}_{\beta\circ n}$ for general $\beta$, bottom: $\mathcal{I}_{\frac{p}{q}\circ kq}$ where $\lfloor\frac{p}{q}\rfloor=m$. Note that the red intervals are the only ones with a non-full branch.
• Figure 3: The map $T_{\beta\circ n}$ on the left and $T_{n\circ \beta}$ on the right for $\beta=\frac{7}{3}$ and $n=3$.
• Figure 4:
• Figure 5: The map $T_{\beta_1}\circ T_{\beta_2}$ on the left and $T_{\beta_3}\circ T_{\beta_4}$ on the right for Example \ref{['ex:nonequalmeasures']}.

Theorems & Definitions (22)

• Theorem 1.1: HW25
• Theorem 1.2
• Proposition 2.1
• proof
• Remark 2.2
• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3