Invariant of topological pressure under some semi-conjugates (Q1370305)

Let \(X,\overline X\) be compact metric spaces, \(f:X\to X\), \(\overline f:\overline X\to\overline X\) continuous maps and \(p:\overline X\to X\) a continuous surjection such that \(p\circ\overline f=f\circ p\). Let \(\varphi:X\to\mathbb{R}\) be a continuous map. In the paper it is shown that the topological pressure \(P(f,\varphi)\) equals the topological pressure \(P(\overline f,\varphi\circ p)\) if one of the following assumptions is satisfied: either \(\overline X\) is a covering space of \(X\) and \(p\) is the covering projection or \((\overline X,\overline f)\) is the inverse limit of \((X,f)\) and \(p\) is the corresponding projection. The result is then applied to obtain a formula for topological entropy of positively expansive maps satisfying specification.