The equivalence of multifractal measures on cookie-cutter-like sets

In this paper, we concentrate on the properties of the multifractal centered Hausdorff measure and the multifractal packing measure on a class of cookie-cutter-like (CCL) sets. We conclude that these two multifractal measures are equivalent on the CCL sets satisfying the strong separation condition. Then we obtain some relevant conclusions about the image measure of a σ-invariant ergodic Borel probability measure.