The variational principle for the defect of factor maps (Q1293991)

This paper deals with the defect of a factor map \(\pi: (Y,\psi)\to (X,\varphi)\), between dynamical systems with \(Y\) totally disconnected. The defect factor is a number \(D(\pi)\in [0,\infty)\) which gives a numerical indication of how far \(\pi\) is from being a conjugacy. In the case, when \(Y\) is metrizable, in his previous paper the author proved \[ D(\pi)\geq \sup_\mu \int_X \log \# \pi^{-1}(x) d\mu(x), \tag{1} \] where \(\mu\) varies over all \(\varphi\)-invariant Borel probability measures on \(X\), and that equality holds in (1) when there is only one \(\mu\) to consider, that is when \(\varphi\) is uniquely ergodic. The main goal of this work is to remove this assumption on \(\varphi\) and prove that equality always holds in (1). Hence, this identity can be considered as a variational principle for the defect.