Let K be an uncountable compact metric space. We prove that a prevalent f∈C(K,Rd) has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that a prevalent f∈C(K,Rd) has graph of maximal Hausdorff dimension, generalizing a result of Bayart and Heurteaux. We obtain similar results for the packing dimension.
We show that for a prevalent f∈C([0,1]m,Rd) the set of y∈f([0,1]m) for which dimHf−1(y)=m contains a dense open set having full measure with respect to the occupation measure λm∘f−1, where dimH and λm denote the Hausdorff dimension and the m -dimensional Lebesgue measure, respectively. We also prove an analogous result when [0,1]m is replaced by any self-similar set satisfying the open set condition.
We cannot replace the occupation measure with Lebesgue measure in the above statement: We show that the functions f∈C[0,1] for which positively many level sets are singletons form a non-shy set in C[0,1]. In order to do so, we generalize a theorem of Antunović, Burdzy, Peres and Ruscher. As a complementary result we prove that the functions f∈C[0,1] for which dimHf−1(y)=1 for all y∈(minf,maxf) form a non-shy set in C[0,1].
We also prove sharper results in which large Hausdorff dimension is replaced by positive measure with respect to generalized Hausdorff measures, which answers a problem of Fraser and Hyde.