Existence of an invariant ergodic measure for infinite parsing collections

Establish whether, when the prefix-free collection of words V is infinite and the associated parsing space X is not necessarily compact, there always exists an ergodic probability measure n on X relative to the induced transformation.

Background

The paper associates to a prefix-free collection V of words in a finite alphabet a dynamical space X consisting of compatible parsings, together with an induced transformation and a factor map onto the original stationary process. For finite V, compactness enables the existence of an ergodic invariant probability measure n on X whose factor is the given stationary process.

If V is infinite, the parsing space X need not be compact. The authors explicitly ask whether an ergodic probability measure with the required invariance properties is nevertheless guaranteed to exist in this noncompact setting. Such a measure would support the use of ergodic-theoretic arguments for parsing spaces generated by infinite word collections.