The box dimension of completely invariant subsets for expanding piecewise monotonic transformations
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.
expanding piecewise monotonic transformation
scientific article; zbMATH DE number 874080