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Common invariant measures for beta-transformations with rational affine relations
We classify common invariant probabilities for two beta-transformations with a rational affine relation between their bases. Let $β>1$ be irrational and $γ=aβ+c>1$, where $a,c\in\mathbb Q$ and $a+c\ne1$. Every finite nonnegative measure that is singular with respect to Lebesgue measure and invariant
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