On normal numbers and self-similar measures (Q2118922)

Let \(\{f_i(x)=s_i\cdot x+t+i\}\) be a self-similar iterated function system on \(\mathbb R\), where \(s_i\in(-1,1)\backslash\{0\}\) and \(t_i\in\mathbb R\), and let \(\beta>1\) be a Pisot number. In this paper it is proved that if \(\log|s_i|/\log\beta\not\in\mathbb Q\) for some \(i\) then for every \(C^1\) diffeomorphism \(g\) and every non-atomic self-similar measure \(\mu\), the measure \(g\mu\) is supported on numbers that are normal in base \(\beta\).