Entropy of random substitution systems
Title: Entropy of random substitution systems
Abstract: Random substitutions are generalisations of substitutions,
where letters are mapped randomly and independently to one of a finite
set of possible words. This typically gives rise to dynamical systems
with a hierarchical structure, mixed spectral type, and a higher
complexity than classical substitution systems. In fact, random
substitution systems usually have positive entropy.
Among the many ergodic measures, a special role is played by the
measures that maximise the entropy – if there is a unique such measure
the system is called intrinsically ergodic. We show that for certain
random substitution systems, the measures of maximal entropy are
precisely those that are invariant under the so called shuffle group,
introduced in previous work of Fokkink-Rust-Salo. This leads to an
equivalent criterion for intrinsic ergodicity in terms of an associated
Markov chain. Finally, we illustrate the richness of this class by
providing an example with several measures of maximal entropy (joint
work with A. Mitchell).