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On balanced subshifts and the existence of invariant Gibbs measures Jung, Uijin


It is well known that if $f$ is a Holder continuous function from a mixing shift of finite type $X$ to $\mathbb R$, then there exists a unique equilibrium state which is an invariant Gibbs measure having $f$ as a potential function. This result has been generalized to wider classes, such as when $X$ is a subshift with the specification property and $f$ is a function in the Bowen class. Recently Baker and Ghenciu showed that there exists a (non-invariant) Gibbs measures for the zero potential if and only if $X$ is (right-)balanced. We extend this result and show that a necessary and sufficient condition for the existence of invariant Gibbs measures on $X$ for the potential $0$ is the bi-balanced condition for $X$. We define a new condition, called $f$-balanced condition for the pair $(X,f)$ and present a similar result for the existence of Gibbs measure with respect to $f$. Using this result, we construct a class of shift spaces which have a Gibbs measure but do not have invariant Gibbs measures for the potential $0$, or equivalently, which are one-sided balanced but not bi-balanced, answering a question raised by Baker and Ghenciu.

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