Topological pressure and fractal dimensions of cookie-cutter-like sets

The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter set it is shown that the topological pressure function exists, and that the fractal dimensions such as the Hausdorff dimension, the packing dimension and the box-counting dimension are all equal to the unique zero $h$ of the pressure function. Moreover, it is shown that the $h$-dimensional Hausdorff measure and the $h$-dimensional packing measure are finite and positive.