A sequential view of self--similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension

Michael Coons, James Evans

备注：: 8 pages

英文摘要：

We show that missing $q$-ary digit sets $F\subseteq[0,1]$ have corresponding naturally associated countable binary $q$-automatic sequence $f$. Using this correspondence, we show that the Hausdorff dimension of $F$ is equal to the base-$q$ logarithm of the Mahler eigenvalue of $f$. In addition, we demonstrate that the standard mass distribution $ν_F$ supported on $F$ is equal to the ghost measure $μ_f$ of $f$.

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