The box dimension of completely invariant subsets for expanding piecewise monotonic transformations (Q1912215)

Let \(T\) be an expanding piecewise monotonic transformation of the unit interval, having a regular derivative \(T'\), and let \(A\) be a closed completely invariant subset of \(([0, 1], T)\) which is topologically transitive with positive entropy. The author shows that the unique zero \(z_A\) of the pressure function \(t\mapsto p(T|A,- t\log |T'|)\) is equal to the box, the Hausdorff as well as the packing dimension of \(A\). The assumption that \(T\) is expanding cannot be weakened. There are some analogues to Julia sets.