Dimension and measure for generic continuous images

Abstract

Abstract. We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, X, into R n . The key question is 'what is the generic dimension of f (X)?' and we consider two different approaches to answering it: Baire category and prevalence.In the Baire category setting we prove that typically the packing and upper box dimensions are as large as possible, n, but find that the behaviour of the Hausdorff, lower box and topological dimensions is considerably more subtle. In fact, they are typically equal to the minimum of n and the topological dimension of X. We also study the typical Hausdorff and packing measures of f (X) and, in particular, give necessary and sufficient conditions for them to be zero, positive and finite, or infinite. It is interesting to compare the Baire category results with results in the prevalence setting. As such we also discuss a result of Dougherty on the prevalent topological dimension of f (X) and give some simple applications concerning the prevalent dimensions of graphs of real-valued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux.

Dimensions of graphs of prevalent continuous maps

“…A lower estimate for the Hausdorff dimension of graph(X + f ) is given there, where X : K → R is a fractional Brownian motion restricted to K ⊂ R m and f ∈ C(K, R) is a continuous drift. In fact, the proof easily extends to vector valued functions, and (as pointed out in [3]) Dougherty's result on images handles the case dim H K = 0, see Theorem 5.1. These yield the following theorem.…”

“…, i k ) ∈ I k , where max ∅ = 0 by convention. Then (3) yields that there is a constant c 4 which depends only on s, t, d such that…”

Dimensions of fibers of generic continuous maps

“…The following lemma is standard, see e.g. [5,Lemma 3.8]. As Tietze's extension theorem holds in R n , Lemma 2.12 implies the following.…”

“…In the case of graphs see the papers of Mauldin and Williams [23], Balka, Buczolich, and Elekes [3] for Hausdorff dimension, Humke and Petruska [13], Liu, Tan, and Wu [20] for packing dimension, and Hyde, Laschos, Olsen, Petrykiewicz, and Shaw [15] for box dimensions. Dimensions of images of generic continuous maps were determined by Balka, Farkas, Fraser, and Hyde [5].…”

On Fourier Analytic Properties of Graphs

“…These two answers are as different as possible and give a good indication of the stark differences in the two theories. Prevalence was formulated by Hunt, Sauer and Yorke in the mid 90s [9] and has been used to study the generic dimensions of graphs of continuous functions on numerous occasions, see [3,4,6,7,18,2]. The most general result to date is essentially due to Bayart and Heurteaux [3], although their result was slightly extended in [2], and states that for an arbitrary uncountable closed set E ⊂ [0, 1], the Hausdorff dimension of the graph of a prevalent function f ∈ C(E) is as large as possible, namely dim H E + 1.…”