  An error in the proof of Lemma 2 (ii) in [I. Werner, Math. Proc. Camb. Phil.
Soc. 140(2) 333-347 (2006)], which claims the absolute continuity of
dynamically defined measures (DDM), is identified. This undermines the
assertion of the positivity of a DDM which provides a construction for
equilibrium states in [I. Werner, J. Math. Phys. 52 122701 (2011)]. An explicit
lower bound for the DDM appearing there is computed in the case when all maps
of a contractive Markov system (CMS) are contractions, the probability
functions are Dini-continuous and bounded away from zero, and there exists an
equilibrium state of the CMS which is absolutely continuous with respect to the
initial measure. In the case of the contraction only on average, a generalized
construction is shown to provide a positive set function, but it is unknown
whether it gives a measure on the Borel $\sigma$-algebra, and if it did, the
measure would coincide with the original DDM.
