Given a weakly almost additive sequence of continuous functions with bounded variation ${\mathcal {F}}=\{\log f_n\}_{n=1}^{\infty }$ on a subshift X over finitely many symbols, we study properties of a function f on X such that $\lim _{n\to \infty }({1}/{n})\int \log f_n\,d\mu =\int f\,d\mu $ for every invariant measure $\mu $ on X. Under some conditions, we construct a function f on X explicitly, and study a relation between the property of ${\mathcal {F}}$ and some particular types of f. As applications, we study images of Gibbs measures for continuous functions under one-block factor maps. We investigate a relation between the almost additivity of the sequences associated to relative pressure functions and the fiber-wise sub-positive mixing property of a factor map. For a special type of one-block factor maps between shifts of finite type, we study necessary and sufficient conditions for the image of a one-step Markov measure to be a Gibbs measure for a continuous function.

An infinite sequence $\alpha $ over an alphabet $\Sigma $ is $\mu $-distributed with respect to a probability map $\mu $ if, for every finite string w, the limiting frequency of w in $\alpha $ exists and equals $\mu (w)$. We prove the following result for any finite or countably infinite alphabet $\Sigma $: every finite-state selector over $\Sigma $ selects a $\mu $-distributed sequence from every $\mu $-distributed sequence if and only if $\mu $ is induced by a Bernoulli distribution on $\Sigma $, that is, a probability distribution on the alphabet extended to words by taking the product. The primary—and remarkable—consequence of our main result is a complete characterization of the set of probability maps, on finite and infinite alphabets, for which finite-state selection preserves $\mu $-distributedness. As a consequence, the shift-invariant measures $\mu $ on $\Sigma ^{\omega }$, such that any finite-state selector preserves the property of genericity for $\mu $, are exactly the positive Bernoulli measures.

For an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes.