Packing dimensions of fibers of generic maps on homogeneous compact spaces

Abstract Balka established that the packing and Hausdorff dimensions of the fibers of a generic continuous map f : K → R n are governed by the dimension profiles d * n ( K ) , * ∈ { H , P }. In this work we compute the packing based quantities d P n ( K ) and d p n ( K ) for several important classes of compact metric spaces, including all topologically homogeneous spaces, self-similar sets, and the planar Brownian range. As a consequence of Balka’s general theorem, the supremum of the packing dimensions of the fibers of generic maps on these spaces is dim P ( K ) . For the lower packing dimension we obtain the corresponding generic upper bound, while the matching lower bound remains open. We also introduce inductive topological packing and lower packing dimensions and compute them in several model cases, such as products with cubes and Sierpiński-type sets. In a broader framework, we define a general topological set function dimension associated with an arbitrary monotone set function, develop its inductive versions, and analyze their asymptotic behavior. In particular, when the underlying space has infinite topological dimension, we obtain a precise description of the growth of these inductive dimensions.