A note on the 1-prevalence of continuous images with full Hausdorff dimension

TL;DR

This paper extends the concept of prevalence in the space of continuous functions on compact metric spaces, showing that the typical functions have images with full Hausdorff dimension 1, witnessed by simple one-dimensional measures.

Contribution

It introduces the notion of 1-prevalence, demonstrating that the prevalent property of having full Hausdorff dimension images can be witnessed by measures supported on one-dimensional subspaces.

Findings

Prevalent functions have images with Hausdorff dimension 1.

The witness measure can be supported on a one-dimensional subspace.

Simplifies understanding of the prevalence structure in function spaces.

Abstract

We consider the Banach space consisting of real-valued continuous functions on an arbitrary compact metric space. It is known that for a prevalent (in the sense of Hunt, Sauer and Yorke) set of functions the Hausdorff dimension of the image is as large as possible, namely 1. We extend this result by showing that `prevalent' can be replaced by `1-prevalent', i.e. it is possible to \emph{witness} this prevalence using a measure supported on a one dimensional subspace. Such one dimensional measures are called \emph{probes} and their existence indicates that the structure and nature of the prevalence is simpler than if a more complicated `infinite dimensional' witnessing measure has to be used.