Limit sets of automatic sequences. (Q1874466)

The authors study the limit sets of multidimensional automatic sequences. For a finite set with a distinguished element \(s_0\), consider an \(\ell\)-dimensional sequence \(f\) in \(S^{\mathbb{N}^\ell}\). A sequence \((t_n)\in\mathbb{N}^{\mathbb{N}}\) is a scaling sequence, if \(({1\over t_n} X(f,t_n))\) is a Cauchy sequence with respect to the Hausdorff distance, where \[ X(f, t_n)= \{Q_\ell+\underline j\mid\|\underline j\|_\infty< t_n,\, f(\underline j)\neq s_0\} \] with \(0\)-dimensional cube \(Q_\ell\). The limit associated with a scaling sequence is a limit set of \(f\). They show that any automatic sequence has natural scaling sequences, and that a limit set of an automatic sequence is a union of primitive limit sets in some sense.