HAUSDORFF AND PACKING DIMENSION OF FIBERS AND GRAPHS OF PREVALENT CONTINUOUS MAPS

Abstract. The notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say about the Hausdorff and packing dimension of the fibers of prevalent continuous maps? Let K be an uncountable compact metric space. We prove that the preva-lent f ∈ C(K,Rd) has many fibers with almost maximal Hausdorff dimen-sion. This generalizes a theorem of Dougherty and yields that the 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 the prevalent f ∈ C([0, 1]m,Rd) the set of y ∈ f([0, 1]m) for which dimH f −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