Automatic sequences in automatic systems

Abstract: A sequence is called automatic if it can be obtained as a coding of a fixed point of a substitution of constant length. We study the class of automatic systems, that is systems which arise as orbit closures of automatic sequences.

Since there are only countably many automatic sequences, and since automatic systems usually have uncountably many points, it is interesting to study the combinatorial structure of the subset of an automatic system which comprises all of its points which are automatic. We give a dynamical description of this set, which is analogous to the one obtained by Holton and Zamboni for minimal substitutive systems. In particular, we show that automatic sequences in an infinite minimal automatic system correspond to the rationals in the ring of k-adic integers, the maximal connected equicontinuous factor of the system.

As an application, we show that any minimal substitutive system which factors onto an infinite k-automatic system is itself k-automatic. We also state several conjectures which generalise our results to arbitrary substitutive systems, and explain their relation to Cobham-type results (connected with the ones obtained by Durand in 2011).

dynamical systems

Audience: researchers in the topic

Dynamical systems seminar at the Jagiellonian University