Dimension and measure for generic continuous images

TL;DR

This paper investigates the typical dimensions and measures of images of uncountable compact metric spaces under continuous functions, using Baire category and prevalence methods, revealing nuanced behaviors of various dimensions.

Contribution

It provides a comprehensive analysis of the generic dimensions and measures of continuous images, contrasting Baire category and prevalence approaches, and extends existing results on prevalent dimensions.

Findings

Typically, packing and upper box dimensions are equal to n.

Hausdorff, lower box, and topological dimensions are min(n, topological dimension of X).

Conditions for Hausdorff and packing measures to be zero, positive, finite, or infinite.

Abstract

We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, $X$, into $\mathbb{R}^n$. 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.