Measure centre and minimal centre of attraction (Q687914)

Let \(X\) be a compact metrizable space and \(f:X\to X\) be a continuous function. For a Borel set \(E\) let \(P_ x(E)\) be the limit of \[ n^{- 1}\text{card}\{i:f^ i(x)\in E,\;0\leq i<n\} \] as \(n\to\infty\). A closed subset \(E\) is said to be a centre of attraction of a nonempty set \(X_ 0\subset X\) if \(f(E)\subset E\) and \(P_ x(V(E,\varepsilon))=1\) for all \(x\in X_ 0\), where \(V(E,\varepsilon)\) is the closed \(\varepsilon\)- neighborhood of \(E\). The minimum (in the set-theoretic sense) centre of attraction is denoted by \(C_{X_ 0}\). Furthermore, let \(M_ x\) be the set of all limit points of the sequence \[ \left\{n^{-1}\sum^{n-1}_{i=0}\delta_{f^ i(x)}\right\} \] in the space of all probability measures on \(X\). The minimum (with respect to proper inclusions) closed set \(E\) such that \(f(E)\subset E\) and \(m(E)=1\) for all \(m\in\bigcup_{x\in X_ 0}M_ x\) is said to be the measure centre \(M(X_ 0)\) of \(X_ 0\). The main result of the paper establishes the equality \(C_{X_ 0}=M(X_ 0)\).