Measures of maximal relative entropy with full support (Q2884103)

It is a classical result in symbolic dynamics that every irreducible shift of finite type has a unique measure of maximal entropy called the Parry measure. The Parry measure is a fully supported Markov measure. Let \(\pi:X\to Y\) be a factor map between shift spaces. Let \(\nu\) be an invariant measure on \(Y\). An invariant measure \(\mu\) on \(X\) is said to be a measure of maximal relative entropy over \(\nu\) (where the relative entropy of \(\mu\) is given by the entropy difference \(h(\mu)-h(\nu)\)) if \(\pi(\mu)=\nu\) and \(h(\mu)\geq h(\mu')\) for all \(\mu'\in\pi^{-1}(\nu)\). In this paper, two main results concerning measures of maximal relative entropy are proved. NEWLINENEWLINENEWLINE Theorem 1. Let \(X\) be an irreducible shift of finite type, \(Y\) and \(\pi\) as above, and \(\nu\) a fully supported invariant measure on \(Y\). Then, any measure of maximal relative entropy among the measures in \(\pi^{-1}(\nu)\) is fully supported. This result is proven by bringing additional randomness to a measure \(\mu\) without changing its image \(\nu\), resulting in another measure with an increased entropy. It follows then that this is possible if \(\mu\) is not fully supported. NEWLINENEWLINENEWLINE Theorem 2. Let \(X\), \(Y\) and \(\pi\) be as in Theorem 1. Then for any fully supported invariant measure \(\nu\) on \(Y\), there is a fully supported invariant measure \(\mu\) on \(X\) that projects to \(\nu\). If \(\nu\) is ergodic, \(\mu\) can be chosen to be ergodic. The conclusions of Theorems 1 and 2 also hold when \(X\) is only an irreducible sofic shift. Finally, some counterexamples shows that the above results do not extend to general shift spaces.