Frequency of digits for Countable Markov maps without the big image property

There has been work on Hausdorff dimension on sets based on the frequency of digits for continued fraction expansion. This has been generalised to the setting of countable Markov maps by Fan, Jordan, Liao and Rams. However this works only when the Markov map satisfies the big image property. The project will look at the case when this property is not satisfied and make use of work by Iommi, Todd and Velozo. With his work it should be possible to give a reasonably complete answer to this problem in that setting.